Topology

Syllabus:

i) METRIC SPACES: Elements of metric space theory. Sequences and Cauchy sequences and the notion of completeness, elementary topological notions for metric spaces i.e.open sets, closed sets, compact sets, connectedness, continuous and uniformly continuous functions on a metric space. The Bolzano - Weirstrass theorem, Supremum and infimum on compact sets, Rn as a metric space.
ii) TOPOLOGICAL SPACES: Definitions and Examples; Bases and sub-bases; Subspace and metric topology; closed sets, limit points and continuous functions; product and quotient topology.
iii) SEPARATION: Countability and Seperation axioms, Normal spaces, Urysohn lemma, Tietze extension theorem.
iv) CONNECTEDNESS AND COMPACTNESS: Connected subspaces of the real line, Compact subspaces of the real line, limit point compactness, local compactness. Tychnoffs theorem. One point compactification.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/Top.html