NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (4/9)

Topology

•    Definition and examples

•    Open and closed sets

•    Subspaces

•    Neighborhoods

•    Limit points, Closure of a set

•    Interior, Exterior and boundary of a set

Bases and Sub-bases

•    Base and sub bases

•    Neighborhood bases

•    First and second axioms of countablility

•    Separable spaces, Lindelöf spaces

•    Continuous functions and homeomorphism

•    Weak topologies, Finite product spaces

Separation Axioms

•    Separation axioms

•    Regular spaces

•    Completely regular spaces

•    Normal spaces

Compact Spaces

•    Compact topological spaces

•    Countably compact spaces

•    Sequentially compact spaces

Connectedness

•    Connected spaces, Disconnected spaces

•    Totally disconnected spaces

•    Components of topological spaces



Section-II (5/9)

Metric Space

•    Review of metric spaces

•    Convergence in metric spaces

•    Complete metric spaces

•    Completeness proofs

•    Dense sets and separable spaces

•    No-where dense sets

•    Baire category theorem

Normed Spaces

•    Normed linear spaces

•    Banach spaces

•    Convex sets

•    Quotient spaces

•    Equivalent norms

•    Linear operators

•    Linear functionals

•    Finite dimensional normed spaces

•    Continuous or bounded linear operators

•    Dual spaces

Inner Product Spaces

•    Definition and examples

•    Orthonormal sets and bases

•    Annihilators, Projections

•    Hilbert space

•    Linear functionals on Hilbert spaces

•    Reflexivity of Hilbert spaces