Introduction to Topology, Fall 2020, Universität Basel
Wednesday, 14.15-16.00 (Zoom)
Assistants: Julia Schneider, Pascal Fong, Shuta Nakajima
- Lecture script
- Klaus Jänich, Topologie. Springer-Verlag Berlin Heidelberg, 2005 link
- A. B. Sossinsky, Topology I. Independent University of Moscow link
Tentative list of topics:
- Allen Hatcher, Algebraic Topology (Chapter 0 and 1). Cambridge University Press, 2002 available online
- Structures and spaces: topological spaces, metric spaces, bases. Position of a point with respect to a set: interior, exterior, boundary.
Continuous maps, homeomorphisms. Topological constructions (disjoint union, Cartesian product, quotient spaces, cone, suspension, and join).
- Topological properties: connectedness, path connectedness, compactness. Separation axioms.
- Alexander's Subbasis Theorem. Tychonoff theorem.
- Elements of algebraic topology. Homotopy. Fundamental groups.
- Depending on time, we shall spend some time in the topological zoo: graphs, simplicial spaces, surfaces (maybe even classification of those).
Topological spaces: definition and basic examples (induced topology, trivial topology,
discrete topology, Euclidean topology). Metric spaces: definition and examples. Metric topology.
Continuous maps of topological spaces.
Base and subbase of topology. Topology generated by base and subbase. Product topological space. Comparison of topologies (finer, coarser, equivalent).
Interior, exterior, and boundary of a set. Giving topology via closed sets. Zariski topology. Closure, limit points. Dense subsets.
Hausdorff spaces. Examples and non-examples. A space is Hausdorff iff the diagonal is closed. Separation axioms T0-T4. Zariski topology on the spectrum
of a commutative ring. Normal and regular spaces.
Normal and regular spaces. Second-countable spaces. Metrizable spaces, Urysohn's metrization theorem (statement). Tychonoff topology, its base and subbase.
Countable product of metrizable spaces is metrizable. Hilbert cube.