Page 4 - DMTH503_TOPOLOGY

SYLLABUS

Topology
Objectives: For some time now, topology has been firmly established as one of basic disciplines of pure mathematics. It's ideas
and methods have transformed large parts of geometry and analysis almost beyond recognition. In this course we will study
not only introduce to new concept and the theorem but also put into old ones like continuous functions. Its influence is evident
in almost every other branch of mathematics.In this course we study an axiomatic development of point set topology,
connectivity, compactness, separability, metrizability and function spaces.

Sr. No. Content

1 Topological Spaces, Basis for Topology, The order Topology, The Product

Topology on X * Y, The Subspace Topology.

2 Closed Sets and Limit Points, Continuous Functions, The Product Topology,
The Metric Topology, The Quotient Topology.

3 Connected Spaces, Connected Subspaces of Real Line, Components and Local
Connectedness,

4 Compact Spaces, Compact Subspaces of Real Line, Limit Point Compactness,

Local Compactness
5 The Count ability Axioms, The Separation Axioms, Normal Spaces, Regular

Spaces, Completely Regular Spaces

6 The Urysohn Lemma, The Urysohn Metrization Theorem, The Tietze Extension
Theorem, The Tychonoff Theorem

7 The Stone-Cech Compactification, Local Finiteness, Paracompactness
8 The Nagata-Smirnov Metrization Theorem, The Smirnov Metrization Theorem

9 Complete Metric Spaces, Compactness in Metric Spaces, Pointwise and

Compact Convergence, Ascoli’s Theorem
10 Baire Spaces, Introduction to Dimension Theory