Topology Qualifying Syllabus

Point-set Topology:

Topological spaces and continuous functions:

Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.

Quotients (Topology)

Homeomorphism invariants:

Separation properties (T0, T1, Hausdorff, regular, normal), countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.

Path Connected T2 (Hausdorff) Compact

Continuous deformations:

Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.

Algebraic topology:

Fundamental groups:

Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.

Covering spaces:

Covering map, Lifting theorems; covering space group action; universal covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal covering; applications to group theory.

Homology:

Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.