1. To introduce students to the mathematical notions of space and continuity.
2. To develop students’ ability to reason in an axiomatic framework.
3. To provide students with a foundation for further study in the area of topology and geometry, both within their degree and subsequently.
4. To introduce students to some basic constructions in topological data analysis.
5. To enhance students’ understanding of mathematics met elsewhere within their degree (in particular real and complex analysis, partial orders, groups) by placing it within a broader context.
6. To deepen students’ understanding of mathematical objects commonly discussed in popular and recreational mathematics (e.g. Cantor sets, space-filling curves, real surfaces).

(BH1) An understanding of the ubiquity of topological spaces within mathematics.

(BH2) Knowledge of a wide range of examples of topological spaces, and of their basic properties.

(BH3) The ability to construct proofs of, or counter-examples to, simple statements about topological spaces and continuous maps.

(BH4) The ability to decide if a (simple) space is connected and/or compact.

(BH5) The ability to construct the Cech and Vietoris-Rips complexes of a point set in Euclidean spac. e

(BH6) The ability to compute the fundamental group of a (simple) space, and to use it to distinguish spaces.

Topological spaces. Examples including Euclidean space, discrete and indiscrete spaces, Alexandrov space of a poset.

Continuous maps. Examples including continuous real functions, limits of real sequences, maps of posets.

Homeomorphisms. Subspace and quotient topologies. Examples including real surfaces (as subspaces of Euclidean space and as quotients of polygons), graphs and the Cantor set.

Simplicial complexes. Cech and Vietoris-Rips complexes of a point set in Euclidean space.

Connectedness. Continuous image of a connected space is connected. Intermediate Value Theorem and Brouwer Fixed Point Theorem in dimension 1. The Cantor set is totally disconnected.

Brief discussion of space-filling curves.

Compactness. Continuous image of a compact space is compact. Compact subsets of Euclidean space and the Boundedness Theorem. Compactness of the Cantor set.

Paths and homotopies. Fundamental group. Examples including the fundamental group of the circle. Cayley graphs and simplicial complexes with prescribed fundamental group. Functoriality of the fundamental group. Brouwer Fixed Point Theorem in dimension 2. The Fundamental Theorem of Algebra.