6TOPO Topology, winter semester 2023/2024
A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours if not specified otherwise; tutorials
were running separately)
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).
6TOPO Topology, winter semester 2022/2023
A BRIEF SYNOPSIS
Each class lasted approximately 3 hours, unless stated otherwise.
Class 1 September 19, 2022
Organizational details. The subject of topology. Definition of a topological
space. Examples and non-examples of topological spaces.
Topologies on a 3-elements set,
number of topologies on a finite set.
The standard topology on \(\mathbb R\).
Closed sets. Clopen sets.
The standard topology does not contain clopen sets except of the trivial ones.
([V], pp. xi-xii, 2, 11-13; Munkres, p.76).
A parody of an episode from the movie
"The Bunker"
featuring clopen sets.
Class 2 September 26, 2022
Examples of (non)closed sets in various topologies.
Cantor set as an example
of a closed set. Definition of topology in terms of closed sets.
Base of a topology, examples. ([V], pp. 13-16).
Class 3 October 3, 2022
Discussion of homeworks 1,2. Criteria for a set to be a base
([V],p.16,3.A,3.B,3.C). Equivalence of definitions of a base in
[V] and in Munkres. The standard topology on \(\mathbb R^2\). Criterion
for two bases to be equivalent. Metric spaces, 0-1 metric.
([V],pp.18-20; Munkres, pp.78-81).
Class 4 October 10, 2022
Discussion of homeworks 3,4. Finer and coarser topologies, criterion for a
topology to be coarser (Munkres, Lemma 13.3). Euclidean metric,
Manhattan metric,
metrics on finite sets. Restriction of a metric to a subspace. Balls and
spheres. Metric topology.
([V],pp.18-20,22).
Class 5 October 31, 2022 (approx. 3h40min)
Metrizability of topological spaces, examples of metrizable
and non-metrizable spaces. Operations on metrics.
Separation axiom. Metric topology always satisfies the separation axiom.
Topological equivalence and metric equivalence of metrics.
If \(\rho\) is a metric, then \(\frac{\rho}{1 + \rho}\) is a topologically
equivalent, but not necessary metrically equavalent, metric.
([V],pp.22-23; Munkres, pp.119-121).
Class 6 November 7, 2022 (approx. 3h30min)
Discussion of homeworks 5-11. Subspace topology, examples. Interior.
([V],pp.27,29; Munkres, pp.88-90).
Class 7 November 14, 2022 (approx. 3h30min)
Discussion of homeworks 12-13. Closure, boundary. Adherent, boundary, and limit
points. Definition of continuous and open (i.e., an image of an open set is
open) maps. \(x \mapsto |x|\) is continuous but not open.
([V],pp.30-31,33,59).
Class 8 November 21, 2022 (approx. 4h)
Discussion of homeworks 14,15. Continuous maps, properties, examples,
equivalent conditions for continuity. Equivalence with the
\(\varepsilon-\delta\) definiton of continuity from analysis in the case of
metric topologies. Homeomorphisms (definition, elementary properties).
([V],pp.59-61,67; Munkres, pp.102-105).
Class 9 November 28, 2022 (approx. 3h15min)
Discussion of homework 16. Homeomorphisms (examples).
([V],pp.68-69,72; Munkres, pp.105-106).
Class 10 December 5, 2002 (approx. 3h30min)
Discussion of homeworks 17-18. \(f: X \to Y\) is continuous iff
\(f(Cl\, A) \subseteq Cl\, f(A)\) for any subset \(A \subseteq X\). Planes with the
same number of punctures are homeomorphic (in the subspace topology of the standard
topology). The product topology.
([V],pp.72,136-137; Munkres, pp.86-87).
Class 11 December 12, 2002 (approx. 3h10min)
Discussion of homeworks 19-21. Compact spaces. Hausdorff spaces. Quotient topology, examples.
([V],pp.97,108-110,141-143,145-148).
HOMEWORKS
6/7TOPO Topology, winter semester 2021/2022
Material not formally covered in the last course:
If \((X,\rho_X)\) and \((Y,\rho_Y)\) are two metric spaces, then
\((X \times Y, \rho)\), where
\(\rho((x_1,y_1),(x_2,y_2)) = \rho_X(x_1,x_2) + \rho_Y(y_1,y_2)\) is a metric
space.
Example of metrics \(\rho_1\), \(\rho_2\) such that \(min(\rho_1,\rho_2)\) is
not a metric.
HOMEWORKS
6TOGE Topology and Geometry, winter semester 2020/2021
A BRIEF SYNOPSIS
Each class lasted approximately 1h30min, unless specified otherwise.
(videos: to watch on youtube, send me a request for
access)
Class 1 October 8, 2020 (approx. 1h50 min)
Organizational details. The subject of topology. Definition of the topological
space. First examples. (According to [V], pp. xi-xii, 2, 11).
youtube video (partially)
Class 2 October 9, 2020
Further examples of topological spaces. Topology of the real line.
(According to [V], pp.11-12).
youtube video
Class 3 October 15, 2020
Topologies on a 3-element set (according to Munkres, p.76).
Topology of the real line (continuation). Further examples and non-examples of
topological spaces (according to [V], p.12).
youtube videos:
part 1
part 2
Class 4 October 16, 2020 (approx. 1h50 min)
Closed sets, characterization of topology in terms of closed sets, examples
(according to [V], pp.13-14).
youtube videos:
part 1
part 2
Class 5 October 22, 2020 (approx. 1h40 min)
Neighborhood, examples ([V], p.14). Bases of topological spaces, examples
([V], p.16).
Class 6 October 23, 2020 (approx. 1h40 min)
Criterion for a set of subsets to be a base ([V], p.16, 3.A).
Standard topology on \(\mathbb R^2\), different bases in it
([V], pp.16-17).
Metric spaces, examples,
Manhattan metric.
Balls and spheres ([V], pp.18-20).
Material for self-study: Cantor set ([V], p. 15 and
wikipedia).
youtube video
Class 7 October 29, 2020
Metric topology ([V], p.22).
youtube video
Class 8 November 5, 2020 (approx. 2 hours)
Metrizability, examples. Equivalent metrics ([V], p.22). Topological
subspace ([V], p.27).
youtube video
Class 9 November 6, 2020 (approx. 1h40min)
If \((X,\rho)\) is a metric, then \((X,\frac{\rho}{1+\rho})\) is a metric
([V], 4.33(1)). Properties of balls in \((X,\frac{\rho}{1+\rho})\).
The metrics \((X,\rho)\) and \((X,\frac{\rho}{1+\rho})\) are (topologically)
equivalent ([V], 4.34), but not necessary (metrically) equivalent (in
the sense of ([V], 4.27).
Examples of subspace topologies ([V], 5.1(1),(2)). An open set in a
subset topology is not necessary open in the ambient topology ([V], 5.D).
youtube video
Class 10 November 12, 2020 (approx. 45min)
Interior, exterior, boundary ([V], Chapter 1, Section 6, pp.29-34).
We started to look at definitions and some examples, but
most part of this section is for self-study.
youtube video
Class 11 November 13, 2020 (approx. 1h40min)
Continuous maps ([V], pp.59-60).
December 2, 2020: Self-study assignment:
Homeomorphisms, [V], pp. 67-75. Try to do at least most of the exercises.
Also consult the same material in Munkres, pp. 105-107.
December 3, 2020: Self-study assignment:
Connectedness, [V], pp. 83-88 and compactness, [V], pp. 108-113. Try to do as
many exercise as you can. Also consult the same material in Munkres,
Chapter 3 (except sections 25,29).
Class 12 December 11, 2020 (approx. 40min)
The meaning of connectdness and compacteness, analogs of these notion from the
real analysis
(Borel-Heine theorem).
Product topology ([V], p.136).
\(X \times Y\) is homeomorphic to \(Y \times X\).
Class 13 December 14, 2020 (approx. 3h30min)
Product topology (continuation): product of metrizable (connected, compact)
spaces is metrizable (connected, compact).
A continuous image of a connected (compact) space is connected (compact).
Quotient topology, examples ([V], pp.142,146,148).
Topological groups ([V], pp.187).
Class 14 December 15, 2020 (approx. 2h30min)
Discussion of homeworks 1-4.
Topological groups (continuation), examples:
\((\mathbb R, +)\), \((\mathbb R^*, \cdot)\), \(GL_n(\mathbb R)\).
Homotopy ([V], pp.208-209).
Class 15 December 17, 2020 (approx. 3h)
Discussion of homeworks 5-9.
Homotopy (continuation), homotopy as equivalence relations, examples
([V], pp.208-209,210). Paths, fundamental group, examples
([V], pp.212-216).
HOMEWORKS
Created: Fri Oct 2 2020
Last modified: Mon Sep 23 18:47:25 CEST 2024