M. Starbird, F. Su, Topology Through Inquiry, MAA Press, 2019
All the books are available in electronic form.
= available in
the university library
A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours, unless specified otherwise;
tutorials are running separately)
Class 1 September 23, 2026
Organizational details. The subject of topology. Definition of a topological
space. Examples of topological spaces: trivial and discrete toplogy,
topologies on 2- and 3-element sets, the standard topology on \(\mathbb R\),
chain topology.
Number of topologies on a finite set.
([V], pp. xi-xii, 2, 11-12; Munkres, p.76)
Class 2 September 29, 2026
Closed sets, definition of topology in terms of closed sets.
Clopen sets.
A parody of an episode from the movie
"The Bunker"
featuring clopen sets. Base of a topology, examples. The standard base of the
standard topology on \(\mathbb R\) can be decreased.
([V], pp.13-14,16)
Class 3 October 1, 2026
Any base of the standard topology on \(\mathbb R\) can be decreased.
Criteria for a family of open sets to be a base ([V], 3.A, 3.B, 3.C).
([V], pp.16-17)
Class 4 October 6, 2026
Topological spaces with small number of bases. Finer and coarser topologies.
Criterion for a topology to be coarser (Munkres, Lemma 13.3). The standard
topology on \(\mathbb R^2\).
([V], pp.16-17; Munkres, p.81).
Class 5 October 8, 2026
Non-existence of a topological space having exactly 3 bases (redo). Metric
spaces. Euclidean metric, Manhattan metric, 0-1 metric. Restriction of a metric
to a subspace. Balls and spheres.
([V], pp.18-20).
Class 6 October 9, 2026
Finite metric spaces. Metric topology. Proof that the metric topology is
well-defined (open balls form a base of the topology). Examples of metric
topologies.
([V], pp.20,22; Munkres, pp.119-120).
