Course Description

This course is a study of topological spaces, which are sets with extra structure. In particular, any geometric object, like Euclidean space, sphere, or a torus can be tought of as a topological space, but the unusual feature in this setup is that if an object can be deformed into another, then the two objects are considered to be the same. Topology is thus useful whenever one wants to consider properties of geometric objects that remain unchanged under deformation. Because of this, topology has found applications in physics, chemistry, biology, data analysis, and many other areas. Topology is also a place where analysis and algebra, two seemingly disjoint subjects of mathematics, come together in unexpected and beautiful ways.

Here is a tentative list of topics we will cover.

• Topological spaces, continuous functions, metric spaces, Tietze Extension Theorem;

• Connectedness and compactness, Heine-Borel Theorem, Tychonoff Theorem;

• Quotient spaces, topological groups, groups acting on spaces;

• Homotopy equivalence, null-homotopy, contractibility, retraction, Brouwer Fixed Point Theorem;

• Separation axioms, normal spaces, Urysohn Lemma;

• Cell complexes, triangulation, Euler characteristic, classification of surfaces;

• Function spaces, compact-open topology;

• Fundamental group;

• Applications such as knot theory and topological data analysis (time permitting).