Introduction to Topology, Fall 2020, Universität Basel

Wednesday, 14.15-16.00 (Zoom)

Assistants: Julia Schneider, Pascal Fong, Shuta Nakajima

Literature:

Additionally: Tentative list of topics:

Lectures

16.09

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.

Video

23.09

Base and subbase of topology. Topology generated by base and subbase. Product topological space. Comparison of topologies (finer, coarser, equivalent).

Video

30.09

Interior, exterior, and boundary of a set. Giving topology via closed sets. Zariski topology. Closure, limit points. Dense subsets.

Video

07.10

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.

Video

14.10

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.

Video

21.10

Urysohn's lemma and Urysohns's metrization theorem (proofs). Every metrizable space is Hausdorff and normal.

Video

28.10

Compact spaces and their properties. Continuous image of compact is compact, continuous bijection is a homeomorphism if the sourse is compact and the target is Hausdorff; compact Hausdorff spaces are normal. Finite intersection property. Alexander subbase theorem and Tychonoff theorem. Application: De Bruijn–Erdős theorem (not to be examined).

Video

4.11

Clopen sets. Connected spaces, connected components, cut points. Path-connectedness, path components. An example of connected but not path-connected space. Local path connectedness.

Video

11.11

Quotient topological space. Examples of homeomorphisms. The n-dimensional torus and the real projective space. Cone and suspension.

Video

18.11

Properties of spaces after taking a quotient; a non-hausdorff quotient of a hausdorff space. Compactification, local compactness. The existence of Alexandroff compactification of a locally compact Hausdorff space. Completion of metric spaces, its existence (sketch of proof). Sequential compactness, its equivalence to compactness for metric spaces.

Video

25.11

Topological manifolds and manifolds with boundary. Topology of the real projective space. Topological groups and Lie groups. Examples: GL(n), U(1), tori. SO(3) is homeomorphic to RP3.

Video

2.12

Deformation retraction. Homotopies and homotopy equivalence, contractible spaces. The fundamental group.

Video

9.12

The fundamental group of the unit circle. Covering spaces, lifting homotopies (statement). Spheres of dimension > 1 are simply connected. Applications: the Fundamental Theorem of Algebra, the Brouwer fixed point theorem, the Hedgehog theorem. Embedding of manifolds: partition of unity (part 1).

Video

16.12

Embedding of compact manifolds in Euclidean spaces (proof). Classification of 1-dimensional manifolds (without proof). The Mobius band and the Klein bottle. Non-orientable surfaces. The projective plane is a Mobius band with a disk attached. Simplicial spaces, triangulations. The Euler characteristic. Classification of compact surfaces (statement).

Video