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PAPER – VIII TOPOLOGY

Unit I
: Infinite sets and the Axiom of Choice. Well-ordered sets – The Maximum Principle – Topological spaces – Basis for a Topology – The Order Topology – Product Topology – Closed sets and Limit Points – Continuous Functions – Metric Topology.

Unit II
: Connectedness and Compactness: Connected Spaces – Connected sets in R – Components and path components – Local connectedness – Compact Spaces – Limit Point Compactness – Urysohn Metrization Theorem.

Unit III
: Countability and Separation Axioms: Countability Axioms – Separation Axioms Urysohn’s Lemma – Urysohn Metrization Theorem. Uni IV: The Tychonoff Theorem – Completely regular spaces – The stone-Cech Compactification - Complete Metric Spaces – Compactness in Metric Spaces – Pointwise and Compact Convergences – The Compact-Open Topology – Ascoli’s Theorem

Unit V
: Homotopy of Paths – The Fundamental Group – Covering Spaces – Fundamental Group: Circle, Punctured Plane, Sn. Text Book: 1. Topology A First Course by James R. Munkres, Prentice Hall of India Private Limited, New Delhi, 2000.

Unit-I: Chapter 1: Sections 1.9 – 1.11 Chapter 2: Sections 2.1 – 2.9

Unit-II: Chapter 3: Sections 3.1 – 3.8

Unit-III: Chapter 4: Sections 4.1 – 4.4

Unit-IV: Chapter 5 & 7 Sections 5.1 – 5.3, 7.1, 7.3 – 7.6.

Unit-V: Chapter 8: Sections 8.1 – 8.6

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