An illustrated introduction to topology and homotopy
type: Item
instance of: scholarly article
zbMATH Open document ID: 1323.55001
publication date: 2013-07-26T00:00:00Z
full work available at URL: https://www.taylorfrancis.com/books/9781482220810
Mathematics Subject Classification ID: 55-01
Mathematics Subject Classification ID: 54-01
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 6191822
author: Sasho Kalajdzievski
MaRDI profile type: Publication
title: An illustrated introduction to topology and homotopy
review text: The book has two parts: The first one deals with general topology, starting with an introduction to set theory and the axiom of choice. The equivalence of the axiom of choice, Zorn's Lemma and the well-ordering theorem is mentioned but without proofs. Then the author does not proceed with general topological spaces but with metric spaces. In Chapter 3 general topological spaces are introduced. In Chapter 4 spaces with special properties, including CW complexes are discussed. Then he deals with product spaces, using two different topologies for infinitely many factors, preparing Tychonoff's theorem and applications. The other topology, the box topology for a product of spaces does not satisfy a Tychonoff theorem (i.e., infinite products of compact spaces need not be compact). In another chapter the separation axioms are treated, together with examples that a \(T_n\) space need not be a \(T_{n+\epsilon}\), \(\epsilon \in \{1/2,1\}\) space. All this enables him finally to deal with the Stone-Čech compactification and the determination of spaces which are subspaces of compact spaces (the completely regular spaces).NEWLINENEWLINEThe title of the second part (``Homotopy'') is somewhat misleading. Although the author introduces the concept of a homotopy, he deals only with the fundamental group; higher homotopy groups are only mentioned without giving any details. So this second part would have more properly come up under the title \textit{Fundamental group and combinatorial group theory}.NEWLINENEWLINEFor this and other reasons, this part appears as a new edition of Seifert-Threfall's famous book (which appeared before what is now called \textit{homotopy theory} was developed). Among other topics, one chapter deals with the Seifert-van Kampen theorem and its applications, and two chapters follow about covering spaces. This part contains objects like the horned sphere, the Borsuk-Ulam theorem (proved only for \(n=2\), like Brouwer's fixed point theorem) and it closes with …
reviewed by: Friedrich Wilhelm Bauer
description: scientific article; zbMATH DE number 6191822