Course Overview: Topology is the study of `spatial' objects. Many key topological concepts were introduced in the Metric Spaces course, such as the open subsets of a metric space, and the continuity of a map between metric spaces. More advanced concepts such as connectedness and compactness were also defined and studied. Unlike in a metric space, there is no notion of distance between points in a topological space. Instead, one keeps track only of the open subsets, but this is enough to define continuity, connectedness and compactness. By dispensing with a metric, the fundamentals of proofs are often clarified and placed in a more general setting.
In the first part of the course, these topological concepts are introduced and studied. In the second part of the course, simplicial complexes are defined; these are spaces that are obtained by gluing together triangles and their higher-dimensional analogues in a suitable way. This is a very general construction: many spaces admit a homeomorphism to a simplicial complex, which is known as a triangulation of the space. At the end of the course, the proof of one of the earliest and most famous theorems in topology is sketched. This is the classification of compact triangulated surfaces.
Learning Outcomes: By the end of the course, a student should be able to understand and construct abstract arguments about topological spaces. Their topological intuition should also be sufficiently well-developed to be able to reason about concrete topological spaces such as surfaces.
Course Synopsis: Axiomatic definition of an abstract topological space in terms of open sets. Basic definitions: closed sets, continuity, homeomorphism, convergent sequences, connectedness and comparison with the corresponding definitions for metric spaces. Examples to include metric spaces (definition of topological equivalence of metric spaces), discrete and indiscrete topologies, cofinite topology. The Hausdorff condition. Subspace topology. [2 lectures]
Accumulation points of sets. Closure of a set. Interior of a set. Continuity if and only if \(f \left( \overline{A}\right) \subseteq \overline{f \left(A\right)}\). [2 lectures]
Basis of a topology. Product topology on a product of two spaces and continuity of projections. [2 lectures]
Compact topological spaces, closed subset of a compact set is compact, compact subset of a Hausdorff space is closed. Product of two compact spaces is compact. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Proof that sequential compactness implies compactness in metric spaces. [2 lectures]
Quotient topology. Quotient maps. Characterisation of when quotient spaces are Hausdorff in terms of saturated sets. Examples, including the torus, Klein bottle and real projective plane. [2 lectures]
Abstract simplicial complexes and their topological realisation. A triangulation of a space. Any compact triangulated surface is homeomorphic to the sphere with \(g\) handles \((g\geqslant 0)\) or the sphere with \(h\) cross-caps \((h\geqslant 1)\). (No proof that these surfaces are not homeomorphic, but a brief informal discussion of Euler characteristic.) [6 lectures]
