Topology. An introduction. (English) Zbl 1301.54002

Waldmann, Stefan

Cham: Springer (ISBN 978-3-319-09679-7/pbk; 978-3-319-09680-3/ebook). xii, 136 p. (2014).

The book is built as a series of lecture notes on topology, a classical and fundamental part of mathematics which should be known by every student, offering a first introduction to this topic. The text is self-contained and enriched with many exercises which enable the students to consolidate the notions discussed in the core of the course. The first four chapters, topological spaces and continuity, construction of topological spaces, convergence in topological spaces and compactness form the core of the book. The author discusses extensively the notions of neighborhoods, interior and closure of subsets, the connectedness properties of topological spaces and separation axioms. Next, he introduces some fundamental constructions such as topological manifolds, which are used in differential geometry, final and initial topologies, intensively used in functional analysis and various quotient constructions, employed in algebraic topology. Convergence of nets, nets and filters, ultrafilters are also discussed in detail. The author also introduces the concepts of compactness and sequential compactness and presents important theorems in general topology e.g. Tikhonov’s theorem on the product of compact spaces and some of its applications. Next, other particular topics are discussed: continuous functions, Urysohn’s lemma, Tietze’s theorem, the Stone-Weierstrass theorem and the Arzela-Ascoli theorem. The last chapter concludes with a detailed discussion of Baire’s theorem and its applications, mainly in functional analysis. An analysis of the discontinuities of functions obtained by a limit process from continuous ones is also included. The book is a valuable asset for self-study for students and can be recommended as a solid basis of a topology course.

Reviewer: Corina Mohorianu (Iaşi)

MSC:

54-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
54A05	Topological spaces and generalizations (closure spaces, etc.)
54B05	Subspaces in general topology
54B15	Quotient spaces, decompositions in general topology
54C05	Continuous maps
54D05	Connected and locally connected spaces (general aspects)
54D30	Compactness
54E35	Metric spaces, metrizability

Keywords:

topological spaces; continuity; convergence; Baire’s theorem