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.
Urysohn's lemma and Urysohns's metrization theorem (proofs). Every metrizable space is Hausdorff and normal.
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).
Clopen sets. Connected spaces, connected components, cut points. Path-connectedness, path components. An example of
connected but not path-connected space. Local path connectedness.
Quotient topological space. Examples of homeomorphisms. The n-dimensional torus and the real projective space. Cone and suspension.
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.
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.
Deformation retraction. Homotopies and homotopy equivalence, contractible spaces. The fundamental group.
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).
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