# [[ Reading ]] ➿ Essential Topology (Springer Undergraduate Mathematics Series) Author Martin D Crossley – Tactical-player.co.uk

I have never seen such a beatiful explanation on continuity and its relations to series and sets Now I understand why, when mathematics is lousily explained,everything seemms to be so hard I recommend strongly this book for someone for self study on topology Hope the author can write on other topics of mathematics. This Book Brings The Most Important Aspects Of Modern Topology Within Reach Of A Second Year Undergraduate Student It Successfully Unites The Most Exciting Aspects Of Modern Topology With Those That Are Most Useful For Research, Leaving Readers Prepared And Motivated For Further Study Written From A Thoroughly Modern Perspective, Every Topic Is Introduced With An Explanation Of Why It Is Being Studied, And A Huge Number Of Examples Provide Further Motivation The Book Is Ideal For Self Study And Assumes Only A Familiarity With The Notion Of Continuity And Basic Algebra It s been a while since I ve attended any math classes Topology was one of the subjects I enjoyed the most while in grad school, and I felt like brushing up for fun Previously, I ve used Munkres and Gemignani for point set topology, and my experience with manifolds came from Milnor, and Guilleman Pollack I found that Munkres got bogged down with too many details and Gemignani, while affordable, was a bit out dated The 2 books on manifolds were relatively useless to me I lacked some foundational material to really make use of them Unfortunately, I ve not seen any algebraic topology, and I d like to remedy that.So, enter Essential Topology This book cuts right to the chase It quickly ties continutity from a calculus perspective to the abstract topological notions of inverse images of open sets You then jump right into some basics of point set topology Unlike most books at this level, you re not bogged down with all the separation axioms and crazy pathological point set topology conuter examples Work thourgh this book, and you ll be doing homotopy and homology in no time Sure, you ll need to brush up on some of the missing details if you re to tackle graduate level analysis and you will in those classes books This book will fast track you to modern day mathematics without the tedious extras. I checked this book out of the library and like the clear exposition, so I decided to buy a copy.What was delivered was the copyright 2010 corrected version, and the print quality is very bad very light type which is not easy to read. The author treats very complicated issues in a simple and comprehensive way.Although, I suggest exemplify with applications of topology in Computational Intelligence This is actually not that bad of a book It is reasonably well motivated and has tons of examples although some are pretty tedious Where it fails is its large number of errors There are tons of minor errors scattered throughout, making the book difficult to read There are also some pretty major mistakes The two page proof of theorem 10.11 is blatantly erroneous, and is the standout example I should note, however, that I have the first printing of the book, so it is easily possible that many of these issues have been resolved in the second printing I also feel that it would make sense to have chapters seven and eight switched, so that the chapter on homotopy groups would follow the chapter on homotopy and the chapter on simplicial homology would follow the chapter covering simplicial complexes The only part of chapter eight which relies on chapter seven is the statement of the Whitehead theorem, which is not proven, at the very end of the chapter Another complaint His English does not always seem grammatically correct to me Maybe I m sheltered living in the eastern US For example, Since the arrows rotate 720 degrees as we go around the circle, so deg f 2 does not sound like a full sentence to me If I had only seen it once or twice I might mistake it for another typo, but this sort of sentence structure is all over the book It really disrupts the flow for me.The first half of this book covers point set topology, the second half algebraic If you want to read this book in full, knowing basic algebra is an absolute must If you have familiarity with, for example, quotient groups, free groups, and the rank nullity theorem from linear algebra you should be fine If you only care about the first half, knowing na ve set theory and basic operations on matrices should suffice.To summarize, the exposition is actually pretty good, but there are too many errors for me to recommend it.IMPORTANT Apparently, a lot of mistakes have been corrected in the most recent printing Please read the comments to this review for details. Good introduction to general topology PhysicalThis book is very well printed and the clear pages are securely bound for a paperback The black white text and graphics are a good size for those who require spectacles H.N.D, Undergraduate, Post graduate This book is designed to help second year undergraduates in a largely math based degree The topic of Topology is the studyof continuous functions Topics coveredIt covers Continuity and Topological spaces, Deconstructionist Topology, Euler numbers, Homotopy Groups and fundamental groups,Simplicial and Singular Homoology, Fibre Bundles Prerequisite study,A firm grasp of the concepts of continuity is pressed home using analytical and abstract algebraic based methods ExamplesThe way new topics, such as rigorous continuity , topological subspaces and Hausdorff properties , are introduced uses context friendly explanations.The proofs are helpful and have some rigor , but yet are not made to scare you off.Then topology is the study of continuous functions and so the ultimate goal of Topology should be to describe all the continuous maps between any given pair of topological spaces Essential Topology , Page 91 An easier explanation runs along the lines by a graphical based explorations of changes in functions over a small value of an independent variable, say of the form f 0,2 R These subtle changes within the function are reflected in the graphical output of the functions Its helpful to see this in this way.Other areas are covered Homology groups can are very hard to calculate , Simplicial Homology which can be thought of as rough approximations to the homotopy groups of a space Essential Topology , page 149 This can be seen as it allows changes to made within a topological group of shapes to form another related yet differing group The study of fundamental groups allows functions to be represented by shapes that allow them to be manipulated to change the form of these functions and there inverses see page 140 And this pushes through to Simplicial Homology Modulo 2 that is easier to handle but harder to rigorously get a hold of at this time of writing A nice thing is selected answers are supplied page 213 216 SummaryI have enjoyed most of this book The later chapters can be much rigorous in there explanations and so demand much time to cope withthe explanations If your course suggests another book then I would follow their advice, but for what its worth its a good book to try as it starts from lower skillbase that some book I have encountered Yet it still worth investigating This book is printed by.co.uk page 226