## Topology Prelim topics

## Topics

Topological spaces: basis for a topology, order topology, product topology, subspace topology.

Closed sets and limit points. Interior, exterior, closure and boundary.

Continuous functions.

Quotient topology. Quotient constructions. Cut and Paste analysis of Torus, projective space, Klein bottle, Mobius band and surfaces via quotient constructions.

- Connectedness, local connectedness, path connectedness, components.
Compactness, limit point compactness, sequential compactness and local compactness. One point compactification construction.

Countability axioms: 1st and 2nd countability, Lindelof spaces, separable spaces.

Separation axioms: Hausdorff,

T1 , regular, normal and completely regular spaces. Urysohn Lemma, Urysohn Metrization Theorem, Tietze Extension Theorem.Manifolds, tangent spaces and derivatives, smooth maps, inverse and implicit function theorems, local immersion and submersion theorems, regular value theorem.

Transversality and intersection. Transversality homotopy theorem, Sard's theorem, Mod-2 intersection and degree theory.

## Main References

- J. R. Munkres, Topology, (2nd edition), Prentice Hall Inc, 2000. (Chapters 1 - 9)
- J. Dugundji, Topology Allyn and Bacon, Boston, 1966.
- S. Willard, General Topology. Addison-Wesley Publishing Company Reading Mass. 1970.
- V. Guillemin and A. Pollack, Differential Topology, American Mathematical Society, 2010