## Introduction to Topology, Fall 2020, Universität Basel

Wednesday, 14.15-16.00 (Zoom)

Assistants: Julia Schneider, Pascal Fong, Shuta Nakajima

### Literature:

- Lecture script
- Klaus Jänich, Topologie. Springer-Verlag Berlin Heidelberg, 2005
*link*
- A. B. Sossinsky, Topology I. Independent University of Moscow
*link*

*Additionally:*
- Allen Hatcher, Algebraic Topology (Chapter 0 and 1). Cambridge University Press, 2002
*available online*

*Tentative list of topics:*
- 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).

### 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