Topology

Timetable

Module aim (aims)

General Topology with the introduction to the theory of homotopy is an essential part of mathematics, necessary to continue the studies in several areas of mathematics, for example analysis, functional analysis, theory of measure and probability. The aim is to provide the basic concepts, fundamental theorems and applications to analysis.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Basics of analysis and elementary set theory.

Syllabus

Week 1 The concept of topology and fundamental facts, examples.Week 2 Compact and connected spaces.Week 3 Tychonoff’s theorem and applications.Week 4 The Stone-Cech compactification I.Week 5 The Stone-Cech compactification II, applications.Week 6 Axioms of separation.Week 7 Urysohn’s theorem, applications.Week 8 Tietze, Dugundji’s extension theorems.Week 9 Locally compact spaces, Alexandrov’s theorem, Tychonoff spaces.Week 10 Topology of spaces Cc(X) and Cp(X).Week 11 Ascoli’s theorem, Stone-Weierstrass theoremWeek 12 Nagata’s theorem about spaces Cp(X).Week 13 Homotopy, homotopic maps.Week 14 Contractible spaces.Week 15 The fundamental group.

Reading list

N. Bourbaki, General Topology, Paris, Hermann 1966R. Engelking, General Topology PWN Warsaw (Polish Edition) 1977J. Kelley, General Topology, Van Nostrand Company 1955.J. Munkres, Topology, Prentice-Hall 1974.W. Rudin, Functional Analysis PWN Warsaw (Polish Edition) 2001I. M. Singer, J. A Thorpe Lecture Notes on Elementary Topology and Geometry, Undegraduate Texts in Mathematics, Springer 1967.