Syllabus

Topological spaces, Basis and subbasis for a topology, Subspace topology, Limit/interior/boundary points, Neighborhood systems, Closure and Interior, Continuous functions, Open and closed maps, Homeomorphism, Product topology, Quotient topology, Metric topology, Order topology.

First and second countable spaces, Separable spaces, T0-spaces, T1-spaces, Hausdorff spaces, Regular spaces, Completely regular spaces, Normal spaces, Urysohn lemma, Tietze extension lemma, Urysohn metrization theorem.

Connected spaces, Connected subspaces of the real line, Components, Path connectedness and local connectedness.

Compact spaces, Compact subspaces of the real line, Tychonoff Theorem, Limit point compactness, Local compactness, Compactification, One-point compactification, Stone-Cech compactification.

To introduce the fundamental concepts of Point-set topology.

To learn general mathematical language via concepts introduced in the course, for higher level analysis courses like Geometric Topology, Algebraic Topology and Differential Topology.

To introduce the notion of homeomorphism, that tells that geometrically different objects may be topologically similar, or may share common topological properties.

To learn about the global and local, structures and properties of topological spaces.

Students will learn about topological spaces, and different ways to introduce topology on a set, via basis, subbasis, closed sets etc.

They will learn about continuous maps, open/closed maps and homeomorphisms. Students will get familiar with the product topology, box topology, the fundamental difference between these two topologies, along with interesting examples related to quotient topology.

Students will learn about countability axioms, separable spaces, metrizable spaces along with hereditary and topological properties.

Students will learn about different separation axioms such as T0, T1, T2 etc. with major characterization theorems such as, Urysohn's lemma, Tietze's extension theorem.

They will be introduced to the notions of local compactness and connectedness, path connectedness and compactification.

J. R. Munkres, Topology, Pearson Prentice Hall

J. L. Kelly, General Topology, Van Nostrand

G. F. Simmons, Introduction to Topology and Modern Analysis, Mac-Graw Hill