General topology

elective courses

The aim of this course is to present a series of main concepts and theorems of general topology which are both important and elegant from the point of view of this field, as well as essential for applications in topology and mathematics as a whole. The notion of compactness and its variants is of central importance to the course.

Basic methods of introducing a topology, weak topologies, Cartesian products, quotient spaces. Separation axioms.

Compactness, Tikhonov theorem. Universality of Tikhonov cubes for classes of completely regular spaces of fixed weight. Compactifications, Alexandrov one-point compactification, Stone-Čech compactification. Stone space of ultrafilters in Boolean algebra. A compact and open topology in the spaces of continuous maps.

Paracompactness, partitions of unity. Paracompactness of metrizable spaces. Metrization theorems (Nagata-Smirnov or Bing).

In addition, the following topics may be discussed:

Elements of descriptive set theory, topological characterizations of the Cantor set, the space of rational numbers, the space of irrational numbers.

Elements of continua theory, local connectedness, local path connectedness, Hahn-Mazurkiewicz theorem.

Michael theorem on continuous selections. Borsuk-Dugundji theorem on the operators of simultaneous extension of continuous functions.

Hyperspace of closed subsets, Vietoris topology, Hausdorff metric.

Elements of the theory of cardinal functions on topological spaces.

A.V. Arkhangel'skii, V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, Reidel, 1984

C. Bessaga, A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975

J. Dugundji, Topology, Allyn and Bacon, 1966

R. Engelking, General topology, Heldermann Verlag, 1989

J. Hocking, G. Young, Topology, Dover Publications, New York 1988

K. Janich, Topology, Springer-Verlag, New York 1980

Student knows the basic methods of introducing topology. Student can use the notions of an infinite Cartesian product of topological spaces and a quotient space. Student knows the axioms of separation.

Student understands the concept of compactness, knows Tikhonov's theorem and Tikhonov's cubes universality theorem. He knows the concept of a compactification and basic constructions of compactifications. Student knows the structure of the Stone space of ultrafilters in Boolean algebra. Student knows the concept of the compact-open topology.

Student knows the concept of a paracompact space, a partition of unity. He can use the theorem on paracompactness of metrizable spaces.

He knows one of the metrization theorems (Nagata-Smirnow or Bing).

exam