Topology

Timetable

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 measures and probability. The aim is to provide the basic concepts, fundamental theorems and applications to analysis.

Module aim (aims)

 Fundamental theorems and applications to analysis.

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

Basics of analysics and elementary set theory.

Syllabus

Week1. The concepts of topology and fundamental facts, examples. Week 2. Compact and connected spaces. Week 3. Tichonoff theorem and applications. Week 4. The Stone-Cech compactification I. Week 5 . The Stone-Cech compactification II. Week 6. Axioms of separation. Week 7. Urysohn's theorem, applications. Week 8. Tietze, Dugungji  extension theorems. Week 9. Locally compact spaces, Alexandrov's theorem, Tichonoff spaces. Week 10. Topology of spaces C_c(X) and C_p(X). Week 11. Ascoli theorem, Stone-Weierstrass theorem. Week 12. Nagata theorem about spaces C_p(X). Week 13. Homotopy, homotopic maps. Week 14. Contractible spaces. Week 15. The fundamental group.

Reading list

N. Bourbaki,  General Topology, Paris 1966, R. Engelking, General Topology, PWN (Warsaw) 1977, J. Kelley, General Topology,  Van Nostrand Company, 1955,  J. Munkres, Topology, Prentice-Hall 1974, W. Rudin,  Functional Analysis PWN 2011, M. Singer, J.  A. Thorpe, Lecture Notes on elementary Topology and Geometry, UndegraduateTexts in Mathematics, Springer  1977.