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海底两万里每一章100字梗概英文回答：Chapter 1: The Mysterious Encounter.In this chapter, the story begins with the introduction of Professor Pierre Aronnax, a renowned French marine biologist. He receives an invitation from the American government to join an expedition to investigate the mysterious sightings of a sea monster. Aronnax, along with his loyal servant Conseil, embarks on a journey aboard the USS Abraham Lincoln, commanded by Captain Farragut. As they search for the creature, they are suddenly attacked by the sea monster. Aronnax, Conseil, and a harpooner named Ned Land are thrown overboard and find themselves clinging to the creature's back. To their surprise, the sea monster turns out to be a highly advanced submarine called the Nautilus, captained by the enigmatic Captain Nemo.Chapter 2: The Underwater World.In this chapter, Aronnax and his companions are taken aboard the Nautilus, where they are introduced to the extraordinary underwater world. Captain Nemo reveals that the Nautilus is powered by electricity and is capable of navigating the depths of the ocean undetected. Aronnax is fascinated by the wonders he witnesses, such as the diverse marine life, underwater forests, and ancient shipwrecks. He also learns about the Nautilus' advanced technology, including its ability to produce fresh air and grow food. As they explore the underwater world, Aronnax realizes that he may never be able to return to the surface and is torn between his desire to explore and his longing for freedom.Chapter 3: The Secret of the Nautilus.In this chapter, Aronnax becomes increasingly curious about Captain Nemo and the secrets of the Nautilus. He learns that Nemo is a genius inventor and scientist who has turned his back on civilization and seeks revenge against those who have wronged him. Aronnax also discovers that Nemo has a deep hatred for war and imperialism, which isreflected in his attacks on warships. Despite his admiration for Nemo's intellect and the wonders of the Nautilus, Aronnax starts to question the morality of Nemo's actions. He struggles to reconcile his admiration for the captain with his own values and beliefs.Chapter 4: The Giant Squid.In this chapter, Aronnax and his companions encounter a giant squid while exploring the depths of the ocean. The Nautilus is attacked by the massive creature, and Captain Nemo decides to engage in a battle with it. Aronnax, Ned Land, and Conseil witness the intense struggle between the Nautilus and the squid, which ends with the squid being defeated. Aronnax is both terrified and amazed by the power of the sea creature and the Nautilus' ability to overcome it. This encounter further deepens Aronnax's fascination with the underwater world and his conflicted feelings towards Captain Nemo.中文回答：第一章，神秘的邂逅。

Chapter1Land and PeopleGreat Britain is the largest island in Europe. It is made up of England, Scotland, and Wales.Together with Northern Ireland, it forms the United Kingdom of Great Britain and Northern lreland. This is the full name of the country which constitutes all these places. The United Kingdom of Great Britain and Northern Ireland, or the United Kingdom, is “the UK" for short. However,most people call the UK ”Britain" or “Great Britain,” and some people simply say “England," which is incorrect and particularly annoys the Scots.According to the 2011 census, the total population of the UK was around 63 million. It is the third-largest in the European Union (behind Germany and France) and the 22nd-largest in the world.The UK is a developed country. According to 2013 statistics it has the sixth-largest national economy in the world (and third-largest in Europe) measured by nominal GDP and eighth-largest in the world (and second-largest in Europe) measured by purchasing power parity (PPP). It was the world's first industrialized country and the world's foremost power during the 19th and early 20th centuries. The UK remains a great power because it still has considerable economic,cultural,military,scientific and political influence internationally.The capital of the UK is London, which is among the world's leading commercial, financial,and cultural centers. Other major cities include Birmingham,Liverpool, and Manchester in England, Belfast and Londonderry in Northern Ireland, Edinburgh and Glasgow in Scotland, and Swansea and Cardiff in Wales.I.Geographical Features1. The UK's Geographical Location and Its SizeThe UK is bordered on the south by the English Channel. It is bordered on the east by the North Sea, and on the west by the Irish Sea and the Atlantic Ocean. The UK's only land border with another nation is between Northern Ireland and Ireland.The UK is separated from the rest of Europe by the English Channel. The English Channel between England and France is quite narrow and the narrowest part is called the Suraits of Dover, which is only 33 km across. In 1985 the British government and the French government decided to build a channel tunnel under the Straits of Dover so that England and France could be joined together by road. After eight years of hard work this channel tunnel, which is called"Chunnel,” was open to traffic in May 1994.The UK covers a total area of 244,110 sq km. lt runs 1,000 km from north to south and extends,at the widest part, about 500 km. So no part of the UK is very far from the coast and it provides a valuable resource. The British coast is long and has good, deep harbors. Sea routes extend far inland, providing cheap transportation.England is the largest, most populous, and wealthiest division of the UK. It makes up 130,400 sq km of the UK's total area.The area of Scotland is 78,800 sq km, the area of Wales is20,800 sq km, and the area of Northern Ireland is 14,100 sq km.This means that England makes up 53.4% of the area of the UK, Scotland 32.3%, Wales 8.5%, and Northern Ireland 5.8%.2.Rivers and LakesSince the UK has a moist climate with much rainfall, it has many rivers and lakes.Rivers in central and eastern Britain tend to flow slowly and steadily all year long because they are fed by the frequent rain. Many have been navigable, and from the earliest times they have served peoples interested in either commerce or invasion. The Highlands act as a divide and determine whether rivers flow west to the Irish Sea or east to the North Sea.Rivers and streams moving westward down from the Highlands tend to be swift and turbulent; rivers flowing eastward tend to be long and gentle, with slowly moving waters.The Thames and the Severn are the longest rivers in Britain and are almost equal in length. The Severn flows south out of the mountains of central Wales to the Bristol Channel at Bristol. It is 354 km long. The Thames,338 km long, flows eastward out of the Cotswold Hills and weaves through the metropolis of London. The Thames provides water to the city of London and is used to carry commercial freight. Other important rivers in England are the Mersey,which enters the Irish Sea at Liverpool; the River Humber on the east coast,into which the Trent River and several other rivers flow; and the Tyne River in northern England,which flows past Newcastle upon Tyne to the North Sea.In Scotland the important rivers are the Clyde and the Forth, which are joined by a canal. The River Clyde flows northwest, past Glasgow, and empties into the Atlantic at the Firth of Clyde. (Firth is the Scottish name for an arm of the sea that serves as the broad estuary of a river.) The River Forth flows eastward into the Firth of Forth, where Edinburgh rises on its south bank.The most important rivers in Northern Ireland are the Lagan, the Bann, and the Foyle.Most of the large lakes in the UK are located in the upland areas of Scotland and northern England,although Lough Neagh in Northern Ireland is the largest lake in the UK. Loch Lomond, on the southwestern edge of the Highlands of Scotland, is the largest on the island of Great Britain, measuring 37 km long and from 1.6 to 8 km ke Windermere is the largest of 15 major lakes in the famous Lake District of northwestern England.It is about 1.6 km wide and more than 16 km long.Ⅱ.Climate1.A Maritime ClimateWhen we say climate we mean the average weather conditions at a certain place over a period of years.We don’t mean the day-to-day weather conditions at a certain place. Though it seems that people are always complaining about the weather in the UK because it is rainy and so changeable and unpredictable,the climate in the UK is in fact a favorable one. The UK has a maritime climate. Winters are mild,not too cold and summers are cool,not too hot.It has a steady reliable rainfall throughout the whole year.It has a small range of temperature.The average temperature in winter in the north is 4—6℃and in summer in the south is 12—17℃.So even in winter one can still see stretches of green grass in the open country,in the parks and round the houses.2.Factors Which Influence the ClimateThe UK is an island country which lies between latitude 50°to 60°north. It lies farther north than even the northernmost par of Heilonjiang Province of China. Compared with other countries of the same latitudes it has a more moderate climate. This is influenced mainly by three factors:(1)The surrounding waters tend to balance the seasonal differences by heating up the land in winter and cooling it off in summer. As the sea heats up and cools off relativelyb slowly it brings warm air in winter and cool air in summer.(2)The prevailing southwest winds or the Westerlies (winds which come from the west) blow over the country all the year round, bringing warm and wet air in winter and keeping the temperatures moderate.(3) The North Atlantic Drift, which is a warm current, passes the western coast of the British Isles and warms them.Since the UK's climate is of the maritime type, it is characterized by cool temperatures, frequent cloudy days and rainstorms. It changes from day to day, and this makes it difficult to forecast. It is so changeable that sometimes one can experience four seasons in the course of a single day. Day may break as a fine spring morning; an hour or so later black clouds may have appeared from nowhere and rain may be pouring down. At midday conditions may really be wintery with the temperatures down to about 8 °C. Then in the Jater atfternoon the sky will be clear, the sun will begin to shine again, and for an hour or two before darkness falls, it will be summer. It has been said that the uncertainty about the weather has had a definite effect upon the Englishman's character. It tends to make him cautious, for example. You may laugh when you see an Englishman going out on a brilliantly sunny morning wearing a raincoat and carrying an umbrella. However, most frequently it comes in drizzles and you don't necessarily need an umbrella.3.RainfallThe UK has a steady reliable rainfall throughout the whole year.The average annual rainfall in the UK is over 1,000 mm. It has 750 mm to 1,250 mm of rainfall along the coast in the east and south except a small area in the southeastern corner of the country which receives less than 750 mm. In the west there is as much as 1,250 mm to 2.000 mm of rainfall and in some areas in the northwest it is over 2,000 mm.The Westerlies blow over the UK all the year round, bringing warm and wet air from the Atlantic Ocean. They rise, climb the highlands and the mountains in the west, become colder and then cause heavy rainfall. There is not so much rainfall in the east because after climbing over the highlands and mountains the air gets warmer and drier when it descends and does not give so much rain.As a result of the rainfall distribution in Britain there is a water surplus in the north and west, and a water deficit in the south and east. Reservoirs have therefore to be built in such highland areas as Central Wales,the Lake District and the Scottish Highlands, so that water can be stored here and then transferred to the more populated and industrial areas of lowland Britain.Generally speaking,the climate in the UK is favorable and equable (neither too hot nor too cold).Extremes of heat or cold, or of drought or prolonged rainfall are rarely experienced. It is estimated that on average about 3-6 cubic meters of rain per person per day fall over the UK. Thisis far more than is needed,but problems still remain. Sometimes there are several months of drought, and at other times too much rain causes flooding. Fog,smog, frost and severe gales are not uncommon and often cause great damage to crops and to people's life. In 1952 the sulphur dioxide in the four-day London smog,an unhealthy atmosphere formed by mixing smoke and dirt with fog, left 4,000 people dead or dying.Since then most cities in the UK have introduced “clean air zones” whereby factories and households are only allowed to burn smokeless fuel.Although the UK does not experience hurricanes, that is,storms with a strong fast wind such as typhoons or cyclones, many areas are subjected to severe gales, especially in winter.III.Plant and Animal Life1.Plant LifeThe mild climate,ample rain, and long growing season in the UK support a great variety of plants,which grow exceptionally well. Most of the UK was once covered with thick,deciduous forests in which oak trees predominated.(Deciduous trees are those that lose their leaves every year.)The impact of centuries of dense human polpulation has massively altered the flora of the UK, and only tiny remnants of these forests remain today.Before they were affected by centuries of clearing and human use, the great oak forests spread over the best soils in Britain. Forests were unable to establish themselves in the poorer soils of the mountains,wetlands,heath,and moorlands.The plants common to these wilder areas are heather,gorse and peat moss.These regions have been altered by heavy grazing of livestock and by controlled burning.Controlled burning creates environments suitable for game birds,which feed on the shoots of the new covered by towns and farmland.The marginal wetlands that remain continue to be threatened by reclamation for farms and homes,and some wetland plant species now grow only in conservation areas.2. Animal LifeBritain has many smaller mammals, and the larger ones tend to be gentle. The only surviving large mammals are red deer, which live in the Scottish Highlands and in Exmoor in southwestern England, and roe deer, foud in the woodlands of Scotland and southern England. At one time boars (wild pigs) and wolves roamed Britain, but they were hunted to extinction.Many smaller mammals inhabit Britain, including foxes,otters, red squirrels, and wildcats. Otters are found mainly in southwestern England and in the Shetland and Orkney islands.The red squirrel, driven out of most of its range by the imported gray squirrel, is now limited mainly to the Isle of Wight and Scotland. Wildcats are found only in parts of Scotland.Bird-watching is a popular national pastime. Britain is home to a large variety of birds, due in lange measure to its position as a focal point of a migratory network.Saltwater fish were once important to Britain's economy. Cod, herring and mackerel are still caught off the coasts of Britain, although quotas are now imposed. Lobster, crab,and other shellfish are caught along inshore waters.IV. People1.Ethnic GroupsFor centuries people have migrated to the British Isles from many parts of the world,some to avoid political or religious persecution, others to find a better way of life or to escape poverty. In historic times migrants from the European mainland joined the indigenous population of Britain during the Roman Empire and during the invasions of the Angles,Saxons,Jutes,Danes,and Normans. The Irish have long made homes in Great Britain.Many Jews arrived in Britain toward the end of the 19th century and in the 1930s. After 1945 large numbers of other European refugees settled in the country. The large immigrant communities from the West Indies and South Asia date from the 1950s and 1960s. There are also substantial groups of Americans, Australians, and Chinese, as well as various other Europeans, such as Greeks, Russians,Poles, Serbs, Estonians, Latvians, Armenians,Turkish Cypriots, Italians, and Spaniards. Beginning in the early 1970s, Ugandan Asians and immigrants from Latin America, Southeast Asia, and Sri Lanka have sought refuge in Britain. People of Indian, Pakistani, and Bangladeshi origin account for more than half of the total ethnic minority population, and people of West Indian origin are the next largest group. The foreign-born element of the population is concentrated in inner-city areas, and more than half live in Greater London.nguagesOf the surviving languages the earliest to arrive in Britain were the two forms of Celtic: the Goidelic (from which Irish, Manx, and Scottish Gaelic derive) and Brythonic (from which the old Cornish language and modern Welsh have developed). Among the contemporary Celtic languages Welsh is the strongest: about one-fifth of the total population of Wales are able to speak it. Scottish Gaelic is strongest among the inhabitants of the islands of the Outer Hebrides and Skye, although it is still heard in the nearby North West Highlands.In Northern Ireland very little Irish is spoken. The last native speakers of Cornish died in the 18th century.The second link with Indo-European is through the ancient Germanic language group, two branches of which, the North Germanic and the West Germanic, were destined to make contributions to the English language. Modern English is derived mainly from the Germanic dialects spoken by the Angles, Saxons, and Jutes (who all arrived in Britain in the 5th century AD) and heavily influenced by the language of the Danes (Vikings), who began raiding the British Isles in about 790 and later colonized parts of northern and eastern England.Under the Norman and Angevin kings, England formed part of a continental empire,and the prolonged connection with France retained by its new rulers and landlords made a deep impression on the English language. Many additions to the English language have been made since the 14th century,but the Normans were the last important linguistic group to enter Britain.3.ReligionThe various Christian denominations in the UK have emerged from schisms that divided the church over the centuries.The greatest of these occurred in England in the 16th century, when Henry VIII rejected the supremacy of the Pope. This break with Rome facilitated the adoption of some Protestant tenets (a principle or belief held by a person) and the founding of the Church of England,still the state church in England,although Roman Catholicism has retained adherents (supporters).In Scotland the Reformation gave rise to the Church of Scotland,which was governed by presbyteries—local bodies composed of ministers and elders—rather than by bishops,as was the case in England.Roman Catholicism in Ireland as a whole was almost undisturbed by theseevents,but in what became Northern Ireland the Anglican and Scottish (Presbyterian) churches had many adherents.The British tradition of religious tolerance has been particularly important since the 1950s, when immigrants began to introduce a great variety of religious beliefs.There are large and growing communities that pactice Islam, Hinduism, and Sikhism.The largest number of Muslims came from Pakistan and Bangladesh.,The lange Sikh and Hindu communities originated in India. There are also many Buddhist groups.4.UrbanizationBy any standard the UK is among the most urbanized of countries. The greatest overall change that accompanied Britain's early industrial development was, in fact, the large-scale urbanization. The increasing percentage of employees in offices and service industries ensures continued urban growth, Of every 10 people in the UK, nine live in towns and more than three of them in one of the country's 10 largest metropolitan areas. The Greater London metropolitan area—the greatest port, the largest center of industry, the most important center of office employment, and the capital city—is by far the largest of these. The need for accommodating business premises (offices) has displaced population from Inner London, and this outward movement, in part, has led to the development of new towns outside the 16-km-wide Green Belt that surrounds London's built-up area.Large metropolitan areas also formed in industrial areas during the 19th and early 20th centuries.Although coalfields or textile manufacture started the initial growth of many of these urban areas, coal mining had virtually ceased in all of them by the end of the 20th century, and heavy industry and textile production had given way to a more diverse form of manufacturing and service activities. Birmingham dominates the extensive built-up area of the West Midlands metropolitan area,but the industrial Black Country—named for its formerly polluted skies and grimy buildings—also has several large and flourishing towns.In Greater Manchester, with a similar number of inhabitants, urbanization accompanied the mechanization of the cotton textile industry. Across the Pennines similar mechanization of wool textiles created the West Yorkshire metropolitan area, with Leeds and Bradford as its twin centers. The metropolitan area of Tyne and Wear (centered on Newcastle upon Tyne) and the Greater Glasgow metropolitan area are also located on coalfields. Greater Glasgow has about one-third of Scotland's people. Merseyside (centered on Liverpool) has traditionally served as a seaport and distribution center for Greater Manchester and the rest of Lancashire.Other large metropolitan areas in Great Britain include South Yorkshire(centered on Sheffield),Nottingham,and Bristol.About one-fifth of Northern Ireland’s people live in Belfast.In addition to these large metropolitan areas,there are many other minor urban areas and large towns,several of which line the coast.With so much urban and suburban concentration, the problems of air,water,and noise pollution have caused much concern in the UK. Clean-air legislation has brought considerable progress in controlling air pollution.Smoke-control areas have been established in most cities and towns, and there has been a shift from coal to cieaner fuels.Pollution of the rivers remains a large problem, particularly in the highly industrialized parts of the UK, but vigilance,research, and control by the National River Authorities and general public concern for the environment are encouraging features of contemporary Britain.5. Population GrowthFrom the 18th century until well into the 19th century, Britain's population soared as the death rate dropped and the birth rate remained high. During this period the total population increased from about 6 million in the 1760s to 26 million in the 1870s. Toward the end of the 19th century and into the 20th century the birth rate stabilized and the death rate remained low. The population took on the characteristics of a modern, developed,and prosperous state. Family size decreased and the median age of the population pared to the rest of the world, the UK has a smaller percentage of younger people and a higher percentage of older people, with more than 20% over the age of 60; those under the age of 15 make up only 13% of the population. Britain's population has been growing slowly, slower than the average for countries in the European Union (EU).6. Migration PatternsBeginning in the 1950s, the immigration of nonwhite(“New Commonwealth") people from such developing nations as India, Pakistan, and the countries of the West Indies became significant, and from 1957 until 1962 there was a net migration gain. Since then restriction on the entry of New Commonwealth citizens has decreased the primary inflow,but dependents of immigrants already in the UK are still admitted. The reasons for restricting entry were in part economic but were also associated with the resistance of the existing population to the new arrivals. Nevertheless, thc UK continues to gain people from the New Commonwealth.Migration within the UK has at times been sizable. Until 1700 the relatively small population was sparsely distributed and largely rural and agricultural, much as it had been in medieval times. From the mid-18th century, scientific and technological innovations created the first modern industrial state.At the same time, agriculture underwent technical and tenurial changes that allowed increased production with a smaller workforce, and revolutionary improvements in transport made the movement of materials and people easier. As a result,by the late 19th century a mainly rural population had largely become a nation of industrial workers and town dwellers.Industry, as well as the urban centers that inevitably grew up around it, concentrated near the coalfields,while the railway network, which grew rapidly ather 1830,enhanced the commercial importance of many towns.The migration of people,especially young people, from the country to industrialized towns took place at an uprecedented rate in the early railway age, and such movements were relatively confined geographically.Migration from agricultural Ireland was an exception, for, when the disastrous potato disease of 1845-1849 led to widespread famine, large numbers moved to Great Britain to become urban workers in Lancashire,Clydeside (the Glasgow region), and London.The rural exodus (a lot of people leaving a place at the same time) went on, but on a greatly reduced scale after 1901.Soon after World War I, new interregional migration flows began when the formerly booming 19th-century industrial and mining districts lost much of their economic momentum. Declining heavy industry in Clydeside, northeastern England, South Wales, and parts of Lancashire and Yorkshire caused a lot of people to lose jobs, and many migrated to the relatively more prosperous Midlands and southern England. This movement of people continued until it was checked by the relatively full employment conditions that occurred soon after the start of World War II.In the 1950s job opportunities in the UK improved with government sponsored diversification of industry, reducing the volume of migration to the south. The decline of certain northern industries—coal mining, shipbuilding, and cotton textiles in particular—had nevertheless reached a critical level by the late 1960s, and the emergence of new growth points in the West Midlands and southeastern England made the drift to the south a continuing feature of British economic life. During the 1960s and 1970s the areas of most rapid growth were East Anglia, the South West, and the East Midlands, partly because of limitations on growth in Greater London and the development of new towns in surrounding areas.During the 1980s the government largely abandoned subsidies for industry and adopted a program of rationalization and privatization. This resulted in the collapse of coal mining and heavy industry in the north and the West Midlands of England and in the Lowlands of Scotland and a similar loss of heavy industry in Northern Ireland, thus creating a wave of migration from these regions to the more prosperous south of England, especially East Anglia,the East Midlands,and the South West.As the economy became stable during the 1990s,migration from Scotland, Northern Ireland, and northern England decreased.While the South East (including Greater London) was the chief destination of external immigrants into Britain, this region, along with the West Midlands, produced a growing internal migration to surrounding regions of England during the 1990s. This pattern reflected a larger trend of migration out of older urban centers throughout Britain to surrounding rural areas and small towns at the end of the 20th century.。

渔夫和他灵魂1至3章读后感英文版全文共6篇示例，供读者参考篇1Title: The Fisherman and His Soul - My ThoughtsIntroduction:Hi everyone! Today, I want to share my thoughts about a really cool book called "The Fisherman and His Soul." I just finished reading the first three chapters, and it was so exciting! Let me tell you all about it.Chapter 1 - The Fisherman's Discovery:In the first chapter, we meet the fisherman. He's akind-hearted person who loves fishing. One day, he catches a beautiful fish, but something magical happens! The fish can talk and asks the fisherman to let it go. I was surprised because I've never heard of a talking fish before! The fisherman decides to release the fish, and they become friends. It made me happy to see the fisherman's kindness.Chapter 2 - The Soul's Adventure:The second chapter was even more fascinating! The fisherman's soul wants to explore the world, so it asks the fisherman if they can separate for a while. I was a little worried because I thought they would be lonely without each other. But the soul promises to come back after one year, so the fisherman agrees. The soul goes on a big adventure, seeing amazing places and meeting many interesting characters. I felt excited for the soul and couldn't wait to find out what would happen next.Chapter 3 - The Soul's Transformation:In the third chapter, the soul returns to the fisherman after a year. But something strange happens! The soul looks different and has turned evil. It wants the fisherman to do bad things, but the fisherman refuses. I was really scared because the soul was acting so mean. It made me think about how important it is to be kind and not let bad influences change us.Conclusion:Reading the first three chapters of "The Fisherman and His Soul" was an amazing experience! I loved following the fisherman's journey and seeing how his soul changed. It taught me about the power of kindness and staying true to ourselves. I can't wait to read the rest of the book and find out what happensnext. If you like adventure and stories about friendship, I highly recommend this book. Happy reading, everyone!Word Count: 237Note: I tried my best to keep the language simple and suitable for a young audience. I hope you find it helpful!篇2The Fisherman and His Soul - A Magical AdventureHave you ever imagined going on an incredible journey through a magical world? Well, in the book "The Fisherman and His Soul" by Oscar Wilde, the main character, a fisherman, embarks on an extraordinary adventure that teaches him valuable lessons about love and sacrifice. I would like to share my thoughts on the first three chapters of this enchanting tale.In the first chapter, we meet the fisherman who falls in love with a beautiful mermaid. Their love is pure and strong, but there is a problem. The fisherman has a human soul, and mermaids don't have souls. They can only gain a soul if they marry a human. This creates a dilemma for the fisherman because he must choose between his love for the mermaid and his soul.As the story unfolds in the second chapter, the fisherman decides to give up his soul for the mermaid. He goes to a witch and asks her to remove his soul. The witch agrees, but warns the fisherman that without a soul, he will lose his conscience and emotions. This part made me think about the importance of our souls. Our souls make us who we are, and without them, we would be empty shells.In the third chapter, the fisherman and the mermaid live happily together, but the fisherman starts feeling empty inside. He realizes that he made a mistake by giving up his soul. Although he loves the mermaid, he misses having emotions and a conscience. This made me understand the significance of having a soul. It's not just about feeling emotions, but also about having a sense of right and wrong."The Fisherman and His Soul" is a captivating story that teaches us about the power of love and the importance of our souls. It reminds us that love sometimes requires sacrifice, but we should never give up the essence of who we are. The fisherman's journey is a metaphor for the choices we make in life. It shows that we should always consider the consequences before making decisions.This book has beautiful descriptions that transport you to a magical world. The author's use of language is poetic and vivid. It made me feel like I was right there with the fisherman, experiencing his emotions and dilemmas. The illustrations in the book also added to the enchanting atmosphere.In conclusion, "The Fisherman and His Soul" is an enchanting tale that takes us on a journey through love, sacrifice, andself-discovery. It reminds us of the importance of our souls and the consequences of our choices. I can't wait to read the rest of the book and find out what happens next. I would recommend this book to anyone who enjoys magical adventures and stories that touch the heart.Remember, my dear friends, always cherish your soul, for it is what makes you truly unique and special.篇3The Fisherman and His Soul - My ThoughtsHi everyone! Today, I want to share my thoughts on a book called "The Fisherman and His Soul." It's a really cool story that I recently read, and I think you'll enjoy it too!In the first chapter, we meet the fisherman. He's a kind and hardworking person who loves the sea. One day, while fishing, he catches a beautiful fish. But guess what? This fish is not an ordinary fish! It has a soul! Can you believe that? I was so surprised when I read that part.The fisherman and the fish become good friends. They talk and sing together. It's like they understand each other without even speaking. They spend a lot of time together, and the fisherman starts falling in love with the fish's soul. He feels so happy when he's with the fish.But here comes the twist! In the second chapter, the fisherman meets a beautiful princess. He falls in love with her instantly. Oh, the power of love! The fisherman wants to marry the princess, but there's a problem. His soul is still connected to the fish's soul. If he wants to be with the princess, he has to get rid of his soul. That's a tough decision to make, right?In the third chapter, the fisherman goes to a witch to ask for help. The witch tells him that if he wants to get rid of his soul, he needs to do something really difficult. He has to go to the edge of the sea and cut off his shadow. Can you imagine cutting off your own shadow? I couldn't believe it when I read that part. It sounded so scary!The fisherman loves the princess so much that he decides to do it. He goes to the edge of the sea, takes a knife, and cuts off his shadow. But as soon as he does that, something terrible happens. The fisherman's soul starts to hurt, and he feels so sad. He realizes that he made a mistake. Love is important, but losing your soul is not worth it.I learned a lot from this book. It made me think about the choices we make and the things we give up for love. It's important to follow our hearts, but we should also remember who we are and what makes us special. The fisherman forgot about his own soul because he was so focused on the princess. That's not a good thing to do."The Fisherman and His Soul" is a magical and exciting story. It teaches us about love, friendship, and the importance of staying true to ourselves. I really enjoyed reading it, and I hope you will too. Let's remember to always keep our souls and be true to who we are.篇4The Fisherman and His Soul - Chapters 1 to 3: My ThoughtsHey everyone! Today I want to tell you about this amazing book I just read called "The Fisherman and His Soul." It's a supercool story written by Oscar Wilde. Let me share with you what I thought about the first three chapters!In the first chapter, we meet the fisherman, who is a very kind and hardworking person. He loves fishing, and one day, he catches a beautiful fish with golden scales. But guess what? The fish can talk! I was so surprised when I read that part. The fisherman and the fish become friends, and the fish promises to grant the fisherman's wishes. Isn't that awesome? I wish I had a talking fish as my friend!In the second chapter, the fisherman makes a wish to be able to breathe underwater, just like the fish. And guess what happens? His wish comes true! Now he can swim and explore the ocean depths. I was so excited reading about all the amazing creatures he saw underwater, like colorful fish and sparkling corals. It made me want to go on an underwater adventure too!But, you know, sometimes wishes can bring trouble. In the third chapter, the fisherman falls in love with a beautiful mermaid. They have such a lovely time together, but there's a catch. The fisherman's soul is trapped inside his body, and he can't be with the mermaid forever. This made me feel sad because they really liked each other. It made me think about how important it is tomake wise choices and consider the consequences of our actions."The Fisherman and His Soul" is teaching me some important lessons. It reminds me that friendship is precious, even if it's with a talking fish. It also shows me that sometimes we have to be careful what we wish for because it might not turn out the way we expect. And most importantly, it tells me that true love is not always easy, and we need to make sacrifices for it.I love how Oscar Wilde writes this story. His words are so beautiful and descriptive. I can imagine the fish swimming gracefully in the ocean and the mermaid's enchanting voice. The story has made me feel so many emotions - happiness, excitement, and even a little bit of sadness. It's like going on a roller coaster ride of feelings!I can't wait to read the next chapters and find out what happens to the fisherman and his soul. Will they find a way to be together? I hope so! I'm learning so much from this book, and I think you would enjoy it too. It's a magical adventure that takes you to a world beyond our own.That's all for now, my friends. I hope you enjoyed my thoughts on "The Fisherman and His Soul." Remember, dreamscan come true, but we should always think about what truly matters to us. See you next time!Word count: 380 words篇5Book Report on The Fisherman and His Soul (Chapters 1-3)Hi everyone! I want to tell you about this really cool bookI've been reading called "The Fisherman and His Soul" by Oscar Wilde. It's a fairy tale but not one of those boring ones with princesses locked in towers. This one has adventure, magic, and even gets kind of spooky!The book starts off with a young fisherman who lives in a tiny hut by the sea. Every night he rows his little boat out and casts his nets to catch fish. One night, as he's pulling in his nets, he feels something really heavy. He thinks maybe he snagged an old chest full of gold and jewels from a sunken pirate ship! But when he finally gets the nets onto the boat, it's not treasure at all. It's a little mermaid whose tail is all tangled up in the nets.At first the fisherman is disappointed it's not gold and jewels. But then he sees how beautiful the mermaid is with her long green hair and sparkling tail. He frees her from the nets and can'ttake his eyes off her. The mermaid tells him she will grant him a wish for saving her life. Without even thinking, the fisherman blurts out "I wish I could love!"You see, the fisherman is so focused on his work that he doesn't have any feelings or a soul. The mermaid says she will give him a human soul so he can experience all the emotions like love, passion, and sorrow. But she warns him that once he has it, there's no going back. The fisherman doesn't even hesitate and agrees to take the soul.When he wakes up the next morning, everything looks completely different. The seagulls' cries sound like beautiful music, the air smells like a million flowers, and the silver and green waves seem to dance for him. He has his soul now and is overwhelmed by all the new sights, smells, and sounds.For the first time, he truly sees the beauty all around him. He even sees his own reflection in a pool and thinks he looks like a young god or prince. But what strikes him most is the vision of a beautiful maiden walking on the beach. The fisherman has fallen in love and spends his days pining for this mystery girl.Whenever she walks on the beach, he calls out to her but she can't hear him. He tries to approach her but then gets too nervous or tongue-tied. The fisherman is so distracted by hisnewfound emotions that he neglects his fishing duties. Some days he doesn't even go out on his boat because he's too busy daydreaming about the girl.Eventually, the maiden disappears and the fisherman is heartbroken. He wanders the beach crying and calling out her name until his soul suddenly separates from his body and takes the form of a young man. The soul starts mocking the fisherman, saying he is weak for obsessing over some human girl. It calls him a slave to his emotions and wants to go out and explore the world.The soul leaves the fisherman's body on the beach and begins its own adventure. It comes across a giant merchant ship and secretly boards it. For three whole days, the soul hides among the ship's cargo, not eating or drinking anything. Finally the ship reaches land and the soul disembarks, free at last.That's where Chapter 3 ends, with the soul heading off on its own journey while the fisherman lays motionless on the beach. I can't wait to see what crazy stuff the soul gets up to now that it's independent. And I feel really bad for the poor fisherman who seems totally devastated at losing his soul. Will he ever get it back or be the same?I'm hooked on this story so far! The writing is beautiful and poetic but also simple enough for me to understand. And the plot reminds me of some other classic fairy tales but with a much deeper meaning behind it. I think Oscar Wilde is trying to say something about the dangers of obsession or being a slave to your emotions. Maybe it's better to live a simple life focused on the present instead of daydreaming all the time.Or maybe the message is that the soul is it's own entity, separate from the body, and shouldn't be caged or suppressed. The fisherman was perfectly content before getting his soul, but then it abandoned him as soon as it tasted freedom. There's so many ways to interpret the symbolism that I can't wait to discuss it with my class!Anyways, I highly recommend checking out this book, especially if you're a fan of folklore, philosophy, or just creative fantasy tales. The story has me completely captivated so far. I'm dying to know what adventures and challenges await the soul as it explores the world. And I'm really hoping the poor fisherman finds a way to get his soul back before something terrible happens to him. Guess I'll have to read on to find out!篇6The Fisherman and His Soul: My ThoughtsHi everyone! Today I want to share with you my thoughts on the book "The Fisherman and His Soul." It's a really interesting story that I recently read, and I hope you enjoy my little review!The book is divided into different chapters, and I want to talk about the first three chapters. Let's dive right in!Chapter 1: The Fisherman's DiscoveryIn the first chapter, we meet the main character, the fisherman. He was a simple man who loved fishing by the sea. One day, while he was fishing, he caught a beautiful fish. But when he tried to let it go, the fish pleaded with him not to. The fish told him that it was not an ordinary fish, but a magical creature called a "Soul." This surprised the fisherman a lot!Chapter 2: The Fisherman's DilemmaIn the second chapter, the fisherman was faced with a dilemma. The Soul wanted to be free and experience the world, but it needed the fisherman's help. The fisherman had to make a difficult decision. Should he keep the Soul with him, or should he set it free? This made me think about how sometimes we have to make tough choices in life.Chapter 3: The Fisherman's JourneyIn the third chapter, the fisherman and the Soul embark on a journey together. They travel to distant lands and encounter many adventures along the way. The fisherman learns a lot from the Soul about love, friendship, and the beauty of the world. This chapter made me feel excited because I love going on adventures too!My Thoughts and FeelingsI really enjoyed reading the first three chapters of "The Fisherman and His Soul." The story is full of magic and wonder, and it made me think about important things in life. I liked how the fisherman and the Soul became friends and supported each other. It reminded me of the importance of friendship and helping others.The language in the book is easy to understand, and the author did a great job of describing the characters and their emotions. The illustrations were also beautiful and helped me imagine the scenes in my mind.One thing I found interesting was how the fisherman had to decide whether to keep the Soul or set it free. It made me think about how sometimes we have to let go of things we love, even if it's difficult, so that they can be truly happy.Overall, I can't wait to read the rest of the book! I want to know what other adventures the fisherman and the Soul will have. If you like magical stories with important lessons, I highly recommend "The Fisherman and His Soul."That's all for now. I hope you enjoyed my review! Happy reading, everyone!。

语言学第一章Chapter 1 Invitation to linguistics1.1 Why study language?1. Language is very essential to human beings.2. In language there are many things we should know.3. For further understanding, we need to study language scientifically.1.2 What is language?Language is a means of verbal communication. It is a system of arbitrary vocal symbols used for human communication.1.3 Design features of languageThe features that define our human languages can be called design features which can distinguish human language from any animal system of communication.1.3.1 ArbitrarinessArbitrariness refers to the fact that the forms of linguistic signs bear no natural relationship to their meanings.1.3.2 DualityDuality refers to the property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level and each of the two levels has its own principles of organization.1.3.3 CreativityCreativity means that language is resourceful because of its duality and its recursiveness.Recursiveness refers to the rule which can be applied repeatedly without any definite limit. The recursive nature of language provides a theoretical basis for the possibility of creating endless sentences.1.3.4 DisplacementDisplacement means that human languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at the moment of conversation.1.4 Origin of language1. The bow-wow theoryIn primitive times people imitated the sounds of the animal calls in the wild environment they lived and speech developed from that.2. The pooh-pooh theoryIn the hard life of our primitive ancestors, they utter instinctive sounds of pains, anger and joy which gradually developed into language.3. The “yo-he-ho” theoryAs primitive people worked together, they produced some rhythmic grunts which gradually developed into chants and then into language.1.5 Functions of languageJacobson——language has six functions:1. Referential指称功能: to convey message and information;2. Poetic: to indulge in language for its own sake;3. Emotive: to express attitudes, feelings and emotions;4. Conative意动功能: to persuade and influence others through commands and entreaties;5. Phatic寒暄功能: to establish communion with others;6. Metalingual元语功能: to clear up intentions, words and meanings.Halliday ——that language has three metafunctions:1. Ideational function: to convey new information, to communicate a content that is unknown to thehearer;2. Interpersonal function: embodying all use of language to express social and personal relationships;3. Textual function: referring to the fact that language has mechanisms to make any stretch of spokenand written discourse into a coherent and unified text and make a living passage different from a random list of sentences.Hu Zhuanglin——language has at least seven functions:1.5.1 InformativeThe informative function means language is the instrument of thought and people often use it to communicate new information.1.5.2 Interpersonal functionThe interpersonal function means people can use language to establish and maintain their status in a society.1.5.3 Performative施为功能The performative function of language is primarily to change the social status of persons, as in marriage ceremonies, the sentencing of criminals, the blessing of children, the naming of a ship at a launching ceremony, and the cursing of enemies.1.5.4 Emotive functionThe emotive function is one of the most powerful uses of language because it is so crucial in changing the emotional status of an audience for or against someone or something.1.5.5 Phatic communionThe phatic communion means people always use some small, seemingly meaningless expressions such as Good morning, God bless you, Nice day,etc., to maintain a comfortable relationship between people without any factual content.1.5.6 Recreational functionThe recreational function means people use language for the sheer joy of using it, suc h as a baby’s babbling or a chanter’s chanting.1.5.7 Metalingual functionThe metalingual function means people can use language to talk about itself. E.g. I can use the word “book” to talk about a book, and I can also use the expression “the word book” t o talk about the sign “b-o-o-k” itself.1.6 What is linguistics?Linguistics is the scientific study of language. It studies not just one language of any one community, but the language of all human beings.1.7 Main branches of linguistics1.7.1 PhoneticsPhonetics is the study of speech sounds, it includes three main areas: articulatory phonetics, acoustic phonetics, and auditory phonetics.1.7.2 PhonologyPhonology studies the rules governing the structure, distribution, and sequencing of speech sounds and the shape of syllables.1.7.3 MorphologyMorphology studies the minimal units of meaning –morphemes and word-formation processes.1.7.4 SyntaxSyntax refers to the rules governing the way words are combined to form sentences in a language, or simply, the study of the formation of sentences.1.7.5 SemanticsSemantics examines how meaning is encoded in a language.1.7.6 PragmaticsPragmatics is the study of meaning in context.1.8 MacrolinguisticsMacrolinguistics is the study of language in all aspects, distinct from microlinguistics, which dealtsolely with the formal aspect of language system.1.8.1 Psycholinguistics心理语言学Psycholinguistics investigates the interrelation of language and mind, in processing and producing utterances (and in language acquisition for example).1.8.2 Sociolinguistics社会语言学Sociolinguistics is the study of the characteristics of language varieties1, the characteristics of their functions2, and the characteristics of their speakers3. (123相互作用构成一个speech community 语言社团)1.8.3 Anthropological linguistics人类语言学Anthropological linguistics studies the history and structure of formerly unwritten language, the emergence of language and the divergence of languages.1.8.4 Computational linguistics计算语言学Computational linguistics is an interdisciplinary field which centers around the use of computers to process or produce human language.1.9 Important distinctions in linguistics1.9.1 Descriptive描写vs. Prescriptive规定A linguistic study is descriptive if it describes and analyses the facts observed.It is prescriptive if it tries to lay down rules for the “correct” use of language.1.9.2 Synchronic共时vs. Diachronic历时The description of a language at some point in time is a synchronic study.The description of a language as it changes through the time is a diachronic study.The distinction between synchronic and dischronic studies is only theoretically clear.原因：①Languages are in a constant state of changing.②The language of any speech community is never uniform.③When a language changes, one set o f features are not suddenly replaced by another set of features.1.9.3 Langue 语言& parole言语Saussure distinguished the linguistic competence of the speaker and the actual phenomena or data of linguistics as langue and parole.Langue refers to the abstract linguistic system shared by all the members of a speech community.Parole refers to the actualized language, or realization of langue.区别：①Langue is abstract but parole is specific to the situation in which it occurs.②Langue is not actually spoken by anyone; parole is alwaysa naturally occurring event.③Langue is relatively stable and systematic; parole is subject to personal and situational constraints.1.9.4 Competence and performanceAccording to Chomsky, a language user’s underlying knowledge about the system of rules is called the linguistic competence, and the actual use of language in concretesituations is called performance.Competence is the ideal language user`s knowledge of the rules of his language.Performance is the actual realization of this knowledge in utterances.。

Inside the Gate走进苗圃园Chapter 1 Waiting for Grandma等待奶奶Today it is dusty outside. The wind is whipping at the earth and sending clouds of brown dirt all around the town.I sit here on the doorstep and watch the women come back from work.Finally, I see Grandma carrying her basket. She is covered in dust and looks tired, but she smiles when she sees me. “What are you up to, Kemzie?” she asks.“Waiting for you!” I say. “I want you to come and help me plant trees.”“Trees?” asks Grandma.I tell Grandma that someone came to speak to us at school today about tree planting.I tell her I want to help out in the nursery just out of town where they grow the trees.“We need the trees to give us wood, shade and fruit, and to help keep the Earth well.” I say quickly. Grandma laughs, and puts her basket down.“You go if you want, but I have to make dinner,” she says.For a moment I feel bad. I shouldn’t leave Grandma now. I should help her chop wood for the fire and cook the food she got at the market.Since Mun and Dad died, Grandma has been looking after all our family. It’s hard work for her.今天外面灰尘很大。

作文梗概爱丽丝漫游奇境记,第一章英文回答：Alice's Adventures in Wonderland, Chapter 1: Down the Rabbit-Hole.Alice, a curious and imaginative young girl, finds herself bored one summer afternoon. As she sits under a tree, she notices a White Rabbit with a pocket watch hurrying by. Intrigued, Alice decides to follow the rabbit down a rabbit-hole.The rabbit-hole is dark and seemingly endless. As Alice falls, she notices shelves lined with jars and books. She also sees a key on a small glass table, but it is out of her reach.Alice continues to fall until she lands in a long hallway. She sees doors of different sizes along the hallway, but they are all locked. Alice finds a smallgolden key on a glass table, and to her delight, it fits into a tiny door. However, she is too large to fit through the door.Alice then discovers a bottle labeled "DRINK ME." Curiosity gets the best of her, and she drinks from it. To her surprise, she shrinks to a tiny size. Now small enough to fit through the door, Alice tries the key again, but it is too small for the lock.Frustrated, Alice begins to cry, and her tears flood the hallway. She notices a cake labeled "EAT ME" and takes a bite. This time, she grows so tall that her head hits the ceiling.Alice's strange journey continues as she encounters a Mouse, a Dodo, and other peculiar creatures. She attends a Caucus-Race, where everyone runs in a circle with no clear winner. Alice also meets the White Rabbit again, who mistakes her for his servant and sends her to fetch his gloves and fan.As Alice ventures further into Wonderland, she encounters more bizarre situations and characters. The chapter ends with Alice finding herself in a room with atiny door, but she has forgotten the key on the table andis unable to open it.中文回答：《爱丽丝漫游奇境记》第一章，进入兔子洞。

Chapter1.Metric Spaces§1.Metric SpacesA metric space is a set X endowed with a metricρ:X×X→[0,∞)that satisfies the following properties for all x,y,and z in X:1.ρ(x,y)=0if and only if x=y,2.ρ(x,y)=ρ(y,x),and3.ρ(x,z)≤ρ(x,y)+ρ(y,z).The third property is called the triangle inequality.We will write(X,ρ)to denote the metric space X endowed with a metricρ.If Y is a subset of X,then the metric space(Y,ρ|Y×Y)is called a subspace of(X,ρ).Example1.Letρ(x,y):=|x−y|for x,y∈I R.Then(I R,ρ)is a metric space.The set I R equipped with this metric is called the real line.Example2.Let I R2:=I R×I R.For x=(x1,x2)∈I R2and y=(y1,y2)∈I R2,defineρ(x,y):=(x1−y1)+(x2−y2).Thenρis a metric on I R2.The set I R2equipped with this metric is called the Euclidean plane.More generally,for k∈I N,the Euclidean k space I R k is the Cartesian product of k copies of I R equipped with the metricρgiven byρ(x,y):=kj=1(x j−y j)21/2,x=(x1,...,x k)and y=(y1,...,y k)∈I R k.Example3.Let X be a nonempty set.For x,y∈X,defineρ(x,y):=1if x=y, 0if x=y.In this case,ρis called the discrete metric on X.Let(X,ρ)be a metric space.For x∈X and r>0,the open ball centered at x∈X with radius r is defined asB r(x):={y∈X:ρ(x,y)<r}.A subset A of X is called an open set if for every x∈A,there exists some r>0 such thatB r(x)⊆A.1Theorem1.1.For a metric space(X,ρ)the following statements are true.1.X and∅are open sets.2.Arbitrary unions of open sets are open sets.3.Finite intersections of open sets are open sets.Proof.Thefirst statement is obviously true.For the second statement,we let(A i)i∈I be a family of open subsets of X and wish to prove that∪i∈I A i is an open set.Suppose x∈∪i∈I A i.Then x∈A ifor some i0∈I.Since A i0is an open set,there exists some r>0such that B r(x)⊆A i.Consequently,B r(x)⊆∪i∈I A i.This shows that∪i∈I A i is an open set.For the third statement,we let{A1,...,A n}be afinite collection of open subsets of X and wish to prove that∩n i=1A i is an open set.Suppose x∈∩n i=1A i.Then x∈A i for every i∈{1,...,n}.For each i∈{1,...,n},there exists r i>0such that B ri(x)⊆A i. Set r:=min{r1,...,r n}.Then r>0and B r(x)⊆∩n i=1A i.This shows that∩n i=1A i is an open set.Let(X,ρ)be a metric space.A subset B of X is called an closed set if its complement B c:=X\B is an open set.The following theorem is an immediate consequence of Theorem1.1.Theorem1.2.For a metric space(X,ρ)the following statements are true.1.X and∅are closed sets.2.Arbitrary intersections of closed sets are closed sets.3.Finite unions of closed sets are closed sets.Let(X,ρ)be a metric space.Given a subset A of X and a point x in X,there are three possibilities:1.There exists some r>0such that B r(x)⊆A.In this case,x is called an interiorpoint of A.2.For any r>0,B r(x)intersects both A and A c.In this case,x is called a boundarypoint of A.3.There exists some r>0such that B r(x)⊆A c.In this case,x is called an exteriorpoint of A.For example,if A is a subset of the real line I R bounded above,then sup A is a boundary point of A.Also,if A is bounded below,then inf A is a boundary point of A.A point x is called a closure point of A if x is either an interior point or a boundary point of A.We denote by A the set of closure points of A.Then A⊆A.The set A is called the closure of A.2Theorem1.3.If A is a subset of a metric space(X,ρ),then A is the smallest closed set that includes A.Proof.Let A be a subset of a metric space.Wefirst show that A is closed.Suppose x/∈A. Then x is an exterior point of A;hence there exists some r>0such that B r(x)⊆A c.If y∈B r(x),thenρ(x,y)<r.Forδ:=r−ρ(x,y)>0,by the triangle inequality we have Bδ(y)⊆B r(x).It follows that Bδ(y)⊆A c.This shows y/∈A.Consequently,B r(x)⊆A c. Therefore,A c is open.In other words,A is closed.Now assume that B is a closed subset of X such that A⊆B.Let x∈B c.Then there exists r>0such that B r(x)⊆B c⊆A c.This shows x∈A c.Hence,B c⊆A c.It follows that A⊆B.Therefore,A is the smallest closed set that includes A.A subset A of a metric space(X,ρ)is said to be dense in X if A=X.§pletenessLet(x n)n=1,2,...be a sequence of elements in a metric space(X,ρ).We say that (x n)n=1,2,...converges to x in X and write lim n→∞x n=x,ifρ(x n,x)=0.limn→∞From the triangle inequality it follows that a sequence in a metric space has at most one limit.Theorem2.1.Let A be a subset of a metric space(X,ρ).Then a point x∈X belongs to A if and only if there exists a sequence(x n)n=1,2,...in A such that lim n→∞x n=x. Proof.If x∈A,then B1/n(x)∩A=∅for every n∈I N.Choose x n∈B1/n(x)∩A for each n∈I N.Thenρ(x n,x)<1/n,and hence lim n→∞x n=x.Suppose x/∈A.Then there exists some r>0such that B r(x)∩A=∅.Consequently, for any sequence(x n)n=1,2,...in A,we haveρ(x n,x)≥r for all n∈I N.Thus,there is no sequence of elements in A that converges to x.A sequence(x n)n=1,2,...in a metric space(X,ρ)is said to be a Cauchy sequence if for any givenε>0there exists a positive integer N such thatm,n>N impliesρ(x m,x n)<ε.Clearly,every convergent sequence is a Cauchy sequence.If a metric space has the property that every Cauchy sequence converges,then the metric space is said to be complete.For example,the real line is a complete metric space.3The diameter of a set A is defined byd(A):=sup{ρ(x,y):x,y∈A}.If d(A)<∞,then A is called a bounded set.Theorem2.2.Let(X,ρ)be a complete metric space.Suppose that(A n)n=1,2,...is a sequence of closed and nonempty subsets of X such that A n+1⊆A n for every n∈I N and lim n→∞d(A n)=0.Then∩∞n=1A n consists of precisely one element.Proof.If x,y∈∩∞n=1A n,then x,y∈A n for every n∈I N.Hence,ρ(x,y)≤d(A n)for all n∈I N.Since lim n→∞ρ(A n)=0,it follows thatρ(x,y)=0,i.e.,x=y.To show∩∞n=1A n=∅,we proceed as follows.Choose x n∈A n for each n∈I N.Since A m⊆A n for m≥n,we haveρ(x m,x n)≤d(A n)for m≥n.This in connection with the assumption lim n→∞d(A n)=0shows that(x n)n=1,2,...is a Cauchy sequence.Since (X,ρ)is complete,there exists x∈X such that lim n→∞x n=x.We have x m∈A n for all=A n.This is true for all n∈I N.Therefore,x∈∩∞n=1A n.m≥n.Hence,x∈A§pactnessLet(X,ρ)be a metric space.A subset A of X is said to be sequentially compact if every sequence in A has a subsequence that converges to a point in A.For example,afinite subset of a metric space is sequentially compact.The real line I R is not sequentially compact.But a bounded closed interval in the real line is sequentially compact.A subset A of a metric space is called totally bounded if,for every r>0,A can be covered byfinitely many open balls of radius r.For example,a bounded subset of the real line is totally bounded.On the other hand, ifρis the discrete metric on an infinite set X,then X is bounded but not totally bounded. Theorem3.1.Let A be a subset of a metric space(X,ρ).Then A is sequentially compact if and only if A is complete and totally bounded.Proof.Suppose that A is sequentially compact.Wefirst show that A is complete.Let (x n)n=1,2,...be a Cauchy sequence in A.Since A is sequentially compact,there exists a )k=1,2,...that converges to a point x in A.For anyε>0,there exists subsequence(x nka positive integer N such thatρ(x m,x n)<ε/2whenever m,n>N.Moreover,there exists some k∈I N such that n k>N andρ(x n,x)<ε/2.Thus,for n>N we havek4ρ(x n,x)≤ρ(x n,x nk )+ρ(x nk,x)<ε.Hence,lim n→∞x n=x.This shows that A iscomplete.Next,if A is not totally bounded,then there exists some r>0such that A cannot be covered byfinitely many open balls of radius r.Choose x1∈A.Suppose x1,...,x n∈A have been chosen.Let x n+1be a point in the nonempty set A\∪n i=1B r(x i).If m,n∈I N and m=n,thenρ(x m,x n)≥r.Therefore,the sequence(x n)n=1,2,...has no convergent subsequence.Thus,if A is sequentially compact,then A is totally bounded.Conversely,suppose that A is complete and totally bounded.Let(x n)n=1,2,...be a sequence of points in A.We shall construct a subsequence of(x n)n=1,2,...that is a Cauchy sequence,so that the subsequence converges to a point in A,by the completeness of A.For this purpose,we construct open balls B k of radius1/k and corresponding infinite subsets I k of I N for k∈I N recursively.Since A is totally bounded,A can be covered byfinitely many balls of radius1.Hence,we can choose a ball B1of radius1such that the set I1:={n∈I N:x n∈B1}is infinite.Suppose that a ball B k of radius1/k and an infinite subset I k of I N have been constructed.Since A is totally bounded,A can be covered by finitely many balls of radius1/(k+1).Hence,we can choose a ball B k+1of radius1/(k+1) such that the set I k+1:={n∈I k:x n∈B k+1}is infinite.Choose n1∈I1.Given n k,choose n k+1∈I k+1such that n k+1>n k.By our construction,I k+1⊆I k for all k∈I N.Therefore,for all i,j≥k,the points x niandx nj are contained in the ball B k of radius1/k.It follows that(x nk)k=1,2,...is a Cauchysequence,as desired.Theorem3.2.A subset of a Euclidean space is sequentially compact if and only if it is closed and bounded.Proof.Let A be a subset of I R k.If A is sequentially compact,then A is totally bounded and complete.In particular,A is bounded.Moreover,as a complete subset of I R k,A is closed.Conversely,suppose A is bounded and closed in I R k.Since I R k is complete and A is closed,A is complete.It is easily seen that a bounded subset of I R k is totally bounded.Let(A i)i∈I be a family of subsets of X.We say that(A i)i∈I is a cover of a subset A of X,if A⊆∪i∈I A i.If a subfamily of(A i)i∈I also covers A,then it is called a subcover. If,in addition,(X,ρ)is a metric space and each A i is an open set,then(A i)i∈I is said to be an open cover.Let(G i)i∈I be an open cover of A.A real numberδ>0is called a Lebesgue number for the cover(G i)i∈I if,for each subset E of A having diameter less thanδ,E⊆G i for5some i∈I.Theorem3.3.Let A be a subset of a metric space(X,ρ).If A is sequentially compact, then there exists a Lebesgue numberδ>0for any open cover of A.Proof.Let(G i)i∈I be an open cover of A.Suppose that there is no Lebesgue number for the cover(G i)i∈I.Then for each n∈I N there exists a subset E n of A having diameter less than1/n such that E n∩G c i=∅for all i∈I.Choose x n∈E n for n∈I N.Since A is sequentially compact,there exists a subsequence(x nk)k=1,2,...which converges to a point x in A.Since(G i)i∈I is a cover of A,x∈G i for some i∈I.But G i is an open set.Hence, there exists some r>0such that B r(x)⊆G i.We canfind a positive integer k such that1/n k<r/2andρ(x nk ,x)<r/2.Let y be a point in E nk.Since x nkalso lies in the setE nk with diameter less than1/n k,we haveρ(x nk,y)<1/n k.Consequently,ρ(x,y)≤ρ(x,x nk)+ρ(x nk,y)<r2+1n k<r.This shows E nk ⊆B r(x)⊆G i.However,E nkwas so chosen that E nk∩G c i=∅.Thiscontradiction proves the existence of a Lebesgue number for the open cover(Gi)i∈I.A subset A of(X,ρ)is said to be compact if each open cover of A possesses afinite subcover of A.If X itself is compact,then(X,ρ)is called a compact metric space. Theorem3.4.Let A be a subset of a metric space(X,ρ).Then A is compact if and only if it is sequentially compact.Proof.If A is not sequentially compact,then A is an infinite set.Moreover,there exists a sequence(x n)n=1,2,...in A having no convergent subsequence.Consequently,for each x∈A,there exists an open ball B x centered at x such that{n∈I N:x n∈B x}is afinite set.Then(B x)x∈A is an open cover of A which does not possess afinite subcover of A. Thus,A is not compact.Now suppose A is sequentially compact.Let(G i)i∈I be an open cover of A.By Theorem3.3,there exists a Lebesgue numberδ>0for the open cover(G i)i∈I.By Theorem 3.1,A is totally bounded.Hence,A is covered by afinite collection{B1,...,B m}of open balls with radius less thanδ/2.For each k∈{1,...,m},the diameter of B k is less thanδ.Hence,B k⊆G ik for some i k∈I.Thus,{G ik:k=1,...,m}is afinite subcover of A.This shows that A is compact.6§4.Continuous FunctionsLet(X,ρ)and(Y,τ)be two metric spaces.A function f from X to Y is said to be continuous at a point a∈X if for everyε>0there existsδ>0(depending onε)such thatτ(f(x),f(a))<εwheneverρ(x,a)<δ.The function f is said to be continuous on X if f is continuous at every point of X.Theorem4.1.For a function f from a metric space(X,ρ)to a metric space(Y,τ),the following statements are equivalent:1.f is continuous on X.2.f−1(G)is an open subset of X whenever G is an open subset of Y.3.If lim n→∞x n=x holds in X,then lim n→∞f(x n)=f(x)holds in Y.4.f(A)⊆f(A)holds for every subset A of X.5.f−1(F)is a closed subset of X whenever F is a closed subset of Y.Proof.1⇒2:Let G be an open subset of Y and a∈f−1(G).Since f(a)∈G and G is open,there exists someε>0such that Bε(f(a))⊆G.By the continuity of f,there exists someδ>0such thatτ(f(x),f(a))<εwheneverρ(x,a)<δ.This shows Bδ(a)⊆f−1(G). Therefore,f−1(G)is an open set.2⇒3:Assume lim n→∞x n=x in X.Forε>0,let V:=Bε(f(x)).In light of statement2,f−1(V)is an open subset of X.Since x∈f−1(V),there exists someδ>0 such that Bδ(x)⊆f−1(V).Then there exists a positive integer N such that x n∈Bδ(x) for all n>N.It follows that f(x n)∈V=Bε(f(x))for all n>N.Consequently, lim n→∞f(x n)=f(x).3⇒4:Let A be a subset of X.If y∈f(A),then there exists x∈A such that y=f(x).Since x∈A,there exists a sequence(x n)n=1,2,...of A such that lim n→∞x n=x. By statement3we have lim n→∞f(x n)=f(x).It follows that y=f(x)∈f(A).This shows f(A)⊆f(A).4⇒5:Let F be a closed subset of Y,and let A:=f−1(F).By statement4we have f(A)⊆⊆F=F.It follows that A⊆f−1(F)=A.Hence,A is a closed subset of X.5⇒1:Let a∈X andε>0.Consider the closed set F:=Y\Bε(f(a)).By statement5,f−1(F)is a closed subset of X.Since a/∈f−1(F),there exists someδ>0 such that Bδ(a)⊆X\f−1(F).Consequently,ρ(x,a)<δimpliesτ(f(x),f(a))<ε.So f is continuous at a.This is true for every point a in X.Hence,f is continuous on X.As an application of Theorem4.1,we prove the Intermediate Value Theorem for continuous functions.7Theorem 4.2.Suppose that a,b ∈I R and a <b .If f is a continuous function from [a,b ]to I R ,then f has the intermediate value property,that is,for any real number d between f (a )and f (b ),there exists c ∈[a,b ]such that f (c )=d .Proof.Without loss of any generality,we may assume that f (a )<d <f (b ).Since the interval (−∞,d ]is a closed set,the set F :=f −1((−∞,d ])={x ∈[a,b ]:f (x )≤d }is closed,by Theorem 4.1.Let c :=sup F .Then c lies in F and hence f (c )≤d .It follows that a ≤c <b .We claim f (c )=d .Indeed,if f (c )<d ,then by the continuity of f we could find r >0such that c <c +r <b and f (c +r )<d .Thus,we would have c +r ∈F and c +r >sup F .This contradiction shows f (c )=d .The following theorem shows that a continuous function maps compact sets to compact sets.Theorem 4.3.Let f be a continuous function from a metric space (X,ρ)to a metric space (Y,τ).If A is a compact subset of X ,then f (A )is compact.Proof.Suppose that (G i )i ∈I is an open cover of f (A ).Since f is continuous,f −1(G i )is open for every i ∈I ,by Theorem 4.1.Hence,(f −1(G i ))i ∈I is an open cover of A .By thecompactness of A ,there exists a finite subset {i 1,...,i m }of I such that A ⊆∪m k =1f−1(G i k ).Consequently,f (A )⊆∪mk =1G i k .This shows that f (A )is compact.Theorem 4.4.Let A be a nonempty compact subset of a metric space (X,ρ).If f is a continuous function from A to the real line I R ,then f is bounded and assumes its maximum and minimum.Proof.By Theorem 4.3,f (A )is a compact set,and so it is bounded and closed.Let t :=inf f (A ).Then t ∈f (A )=f (A ).Hence,t =min f (A )and t =f (a )for some a ∈A .Similarly,Let s :=sup f (A ).Then s ∈f (A )=f (A ).Hence,s =max f (A )and s =f (b )for some b ∈A .A function f from a metric space (X,ρ)to a metric space (Y,τ)is said to be uni-formly continuous on X if for every ε>0there exists δ>0(depending on ε)such that τ(f (x ),f (y ))<εwhenever ρ(x,y )<δ.Clearly,a uniformly continuous function is continuous.A function from (X,ρ)to (Y,τ)is said to be a Lipschitz function if there exists a constant C f such that τ(f (x ),f (y ))≤C f ρ(x,y )for all x,y ∈X .Clearly,a Lipschitz function is uniformly continuous.8Example.Let f and g be the functions from the interval(0,1]to the real line I R given by f(x)=x2and g(x)=1/x,x∈(0,1],respectively.Then f is uniformly continuous, while g is continuous but not uniformly continuous.Theorem4.5.Let f be a continuous function from a metric space(X,ρ)to a metric space(Y,τ).If X is compact,then f is uniformly continuous on X.Proof.Letε>0be given.Since f is continuous,for each x∈X there exists r x>0suchthatτ(f(x),f(y))<ε/2for all y∈B rx (x).Then(B rx(x))x∈X is an open cover of X.Since X is compact,Theorem3.3tells us that there exists a Lebesgue numberδ>0for this open cover.Suppose y,z∈X andρ(y,z)<δ.Then{y,z}⊆B rx(x)for some x∈X. Consequently,τ(f(y),f(z))≤τ(f(y),f(x))+τ(f(x),f(z))<ε/2+ε/2=ε.This shows that f is uniformly continuous on X.9。