Waldmann - Topology: an introduction

Topology: an introduction — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Topology: an introduction" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...112

Springer International Publishing Switzerland 2014

Stefan Waldmann Topology 10.1007/978-3-319-09680-3_1

1. Introduction

Stefan Waldmann 1

(1)

Julius Maximilian University of Wrzburg, Wrzburg, Germany

Stefan Waldmann

Email:

The basic idea of topology is to axiomatize the properties of open subsets in : one can take arbitrary unions and finite intersections of open subsets to obtain again open subsets, and the empty set as well as the total space are open, too. This already provides the precise definition of a topology, i.e. a collection of subsets of a set which should be regarded as open. A key feature of continuous functions on is that a function is continuous iff the inverse images of open subsets are again open. This observation can then be turned into the definition of continuity for maps between topological spaces. With these two definitions one has already the core ingredients to develop the theory of general topological space, called general topology or point set topology.

In these notes we will give some first introduction to topology. The field of topology is a very classical part of mathematics which every student should be exposed to at least once. Many of the notions we will encounter will be helpful if not crucial to understand more advanced topics in mathematics, in particular in functional analysis, differential geometry, and algebraic topology, to name just a few. Nevertheless, these notes provide only an overview and it will remain a task for the reader to continue at other more advanced texts whenever this is needed.

A guiding principle throughout this text will be the situation in which we would like to generalize somehow as far as possible, or at least very far. This way we will obtain topological spaces sharing not too many features with but providing some new and sometimes very unexpected properties. One major difficulty in general topology will be to handle this zoo of new notions and effects in a reasonable way. It will be very important to understand which of the new features of topological spaces occur in many other contexts and have some general interest and which of them provide only rather pathological behaviour. We will only discuss a certain small part of the possible features and their mutual relations.

One of the major advantages of topology is that after passing to the general framework, many theorems become very easy, sometimes almost trivial. Of course, some other difficulties show up and make the whole theory interesting. A good example for the former is the continuity of the composition of two continuous maps, the latter is illustrated e.g. by the complications caused by the difference of continuity using nets or filters and sequential continuity.

The material is presented in several chapters, the first four of them can be seen as the core of the course. After introducing topological spaces and continuous functions we discuss many additional features and properties in Chap. we introduce the concepts of compactness and sequential compactness. Compact spaces usually show a much simpler and nicer behaviour in many ways, so understanding them in quite some detail usually provides a good starting point when moving on to the non-compact case for any sort of problem. One of the most important theorems in general topology is Tikhonovs Theorem stating that a Cartesian product of compact spaces will be again compact. The applications of this theorem are indeed overwhelming and we indicate only few of them in the discussion of Tikhonovs Theorem.

After these four chapters, the following two contain some more particular topics: in Chap. continues with a detailed discussion of Baires Theorem. The notion of meager subsets generalizes the nowhere dense subsets in a very useful way. The statement of Baires Theorem has many applications, mainly in functional analysis. We conclude this short introduction to topology with some rather surprising and amusing facts on the discontinuities of functions which are obtained by limit processes from continuous ones.

In a small appendix we collect some useful formulas from set theory and give a precise formulation of Zorns Lemma and the Axiom of Choice as it is used in the main text. However, this is far from being an introduction to set theoretic concepts but just a cheat sheet instead.

It is clear that such an important and classical field of mathematics like topology can hardly be covered by such a small book. For a more detailed approach and also for several other topics which are not covered in the following, we would like refer to the textbooks [3, 13, 17, 27, 32]. Here one finds additional references, further details and examples as well as many more advanced topics of topology. After mastering the general aspects of topology as presented here, one may want to continue in various directions: algebraic topology including homotopy and sheaf theory is treated in textbooks like [4, 5, 8, 31, 33], differential topology and topological manifolds are discussed in e.g. [12, 22]. The whole world of functional analysis is clearly to vast to describe, but first introductions can be found e.g. in [20, 28, 29, 35] to name only a few.

As a last word of guidance, it should be clear that whenever one wants to learn some topic in mathematics some labour is needed: there are plenty of exercises which help to deepen the understanding of the core text. They also provide some additional examples and open the horizon for further developments of the theory.

Springer International Publishing Switzerland 2014

Stefan Waldmann Topology 10.1007/978-3-319-09680-3_2

2. Topological Spaces and Continuity

Stefan Waldmann 1

(1)

Julius Maximilian University of Wrzburg, Wrzburg, Germany

Stefan Waldmann

Email:

Starting from metric spaces as they are familiar from elementary calculus, one observes that many properties of metric spaces like the notions of continuity and convergence do not depend on the detailed information about the metric: instead, only the coarser knowledge of the set of open subsets is needed. This motivates the definition of a topological space as a set together with a collection of subsets which are declared to be the open subsets . The precise definition requires crucial properties of open subsets as they are valid for metric spaces. Having managed this axiomatization of openness it is fairly easy to transfer the notions of continuity and neighbourhoods to general topological spaces.

Already for metric spaces and now for general topological spaces there are several notions of connectedness which we shall discuss in some detail. New for topological spaces is the need to specify and require separation properties: unlike for a metric spaces we can not necessarily separate different points by open subsets anymore. We will discuss some of these new phenomena in this chapter.

2.1 Metric Spaces

Before defining general topological spaces we recall some basic definitions and results on metric spaces as they should be familiar from undergraduate courses. We start recalling the main definition of a metric space:

Definition 2.1.1

Next page

1234567...112

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «Topology: an introduction»

Look at similar books to Topology: an introduction. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Sasho Kalajdzievski

An Illustrated Introduction to Topology and Homotopy Solutions Manual for Part 1 Topology

Steen

Counterexamples in Topology

Morishita

Knots and primes: an introduction to arithmetic topology

Manetti

Topology

Dover Publications.

Introduction to Topology

Carter

Quantum topology

John B. Conway

A Course in Point Set Topology

Fulton

Algebraic topology: a first course

Stephen Willard

General Topology

Richard Earl

Topology: A Very Short Introduction

H. Graham Flegg

From Geometry to Topology

Bert Mendelson

Introduction to Topology: Third Edition

Discussion, reviews of the book Topology: an introduction and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.