BAISIC CONCEPTS: topological spaces, bases, subbases, closure and interior, convergence of a sequence, continuous maps, subspaces, product spaces, quotient spaces.

CONNECTEDNESS: connected spaces, components, path connected spaces, path components.

COMPACTNESS: compact spaces, Heine-Borel Theorem, the Tychonoff Theorem, local compactness, compactifications, Lindelof spaces, paracompactness.

COUNTABILITY & SEPARATION: the countability axioms, the separation axioms, Urysohn's Lemma, the Tietze Extension Theorem, partitions of unity.

METRIC SPACES: metrics and metric spaces, completion, compactness in metric spaces, Baire spaces and Baire Category Theorem, the Urysohn Metrization Theorem, the Nagata-Smirnov Metrization Theorem.

HOMOTOPY: homotopic maps, contractible spaces, deformation retracts.

FUNDAMENTAL GROUPS: the fundamental group, functorial properties, calculations for surfaces and high dimensional spheres, covering spaces, applications.