## 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

1. J. R. Munkres, Topology, (2nd edition), Prentice Hall Inc, 2000. (Chapters 1 - 9)
2. J. Dugundji, Topology Allyn and Bacon, Boston, 1966.