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.
Reference Texts:
(a) J. Munkres: Topology a first course.
(b) M. A. Armstrong: Basic Topology.
(c) G. F. Simmons: Introduction to Topology and Modern Analysis.
(d) K. Janich: Topology.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/Top.html
- Teacher: Maneesh Thakur