Pasha Zusmanovich - teaching

A.T. Fomenko, Double cover of the Klein bottle by torus

Pasha Zusmanovich

6TOPO Topology, winter semester 2023/2024

Teams

LITERATURE
Main:

J.R. Munkres, Topology, 2nd ed., Prentice Hall, 2000.
O.Ya. Viro, O.A. Ivanov, N.Yu. Netsvetaev, V.M. Kharlamov, Elementary Topology. Problem Textbook, AMS, 2008. Referred in what follows as [V]

Additional:

A.V. Arkhangelskii and V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, D. Reidel, 1984.
R. Engelking, General Topology, revised and completed ed., Heldermann Verlag, 1989.
A. Fomenko and D. Fuchs, Homotopical Topology, 2nd ed., Springer, 2016 (included here solely for its pictures).
M.W. Hirsch, Differential Topology, Springer, 2001.
J.F. Kelley, General Topology, Van Nostrand, 1955; reprinted by Springer, 1975 and Dover, 2017.
C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, 1980. (Russian edition only)
J.W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965.
S.A. Morris, Topology Without Tears.
M. Starbird and F. Su, Topology Through Inquiry, MAA Press, 2019.

All the books are available in electronic form in multiple places.

= available in the university library

A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours if not specified otherwise)

Class 1 September 19, 2023
Organizational details. The subject of topology. Definition of a topological space. Examples of topological spaces: trivial and discrete toplogy, topologies on 0-, 1- and 2-element sets, the standard topology on \(\mathbb R\), the arrow. Number of topologies on a finite set. ([V], pp. xi-xii, 2, 11-12; Munkres, p.76).

Class 2 September 26, 2023
Closed sets. Clopen sets. A parody of an episode from the movie "The Bunker" featuring clopen sets. Cantor set as an example of a closed set. Definition of topology in terms of closed sets. Base of a topology. The standard topology on \(\mathbb R\) does not have a minimal base. ([V], pp.13-16).

Class 3 October 4, 2023
Criteria for a set of subsets to be a base ([V],p.16,3.A,3.B,3.C). Bases of the standard topology on \(\mathbb R^2\). Criterion for two bases to be equivalent. Finer and coarser topologies. ([V], pp.16-17; Munkres, p.81).

Class 4 October 11, 2023
Metric spaces. 0-1 metric, Euclidean metric, Manhattan metric. Restriction of a metric to a subspace. Balls and spheres. ([V], pp.18-20).

Class 5 November 3, 2023
Metric topology. Metrizability of topological spaces, examples of metrizable and non-metrizable spaces. Topological and metric equivalences of metrics. ([V], pp.22-23; Munkres, pp.119-121).

Class 6 November 8, 2023 (approx. 2h)
Metric equivalence implies topological equiavalence. If \(\rho\) is a metric, then \(\frac{\rho}{1+\rho}\) is topologically equivalent, but not necessary metrically equavalent, metric. ([V], pp.22-23; Munkres, pp.119-121).

Class 7 November 15, 2023 (approx. 2h)
Subspace topology, examples. Interior, closure, boundary. Definition of continuous and open (i.e., an image of an open set is open) maps. \(x \mapsto |x|\) is continuous but not open. Equivalent conditions for continuity. ([V], pp.27,29-31,59; Munkres, pp.88-90,102-105).

Class 8 November 22, 2023 (approx. 2h)
Continuity at a point. Equivalence of topological definition of continuity with the \(\varepsilon-\delta\) definiton from analysis in the case of metric topologies. Homeomorphisms (definition, elementary properties). ([V], pp.61,67; Munkres, p.105).

Class 9 November 29, 2023 (approx. 2h40min)
Determining homeomorhisms of various topological spaces (trivial, discerete, cofinite topologies, some finite topology, arrow). Any two intervals on a line, considered as a subspace topologies of the standard topology, are homeomorphic. A semiopen interval is not homeomorphic to an open interval. ([V], pp.68-69).

Class 10 December 6, 2023 (approx. 2h30min)
Homeomorphisms of the arrow (redo). Further examples and non-examples of homeomorphisms. ([V], pp.69-73).

Class 11 December 13, 2023 (approx. 2h)
Homeomorphism and non-homeomorphism of planes with punctures (finishing). The product topology, examples. The quotient topology, examples. ([V], pp.72,136-137,141-143,145-148; Munkres, pp.86-87).

EXAM

Set 2:

Set 3:

A brief synopsis, videos, and homeworks from this course at the previous semesters

Created: Fri Oct 2 2020
Last modified: Wed Feb 28 06:42:23 CET 2024