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).