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Syllabus | Elementary Topology (Code-09050604)

Elementary Topology (Code-09050604)

S.N

Content

Domain

Time                         (Hours)

1

  • Statements only of (Axiom of choice, Zorn’s Lemma, Well ordering theorem and Continuum hypothesis)

Must know

10

  • Definition and examples of topological spaces
  • Neighbourhoods, Interior point and interior of a set
  • Closed set as a complement of an open set
  • Adherent point and limit point of a set,Closure of a set,Derived set
  • Properties of closure operator, Boundary of a set, Dense subsets,Interior,Exterior and boundary operators
  • Base and subbase for a topology
  • Neighbourhood system of a point and its properties
  • Basefor Neighbourhood system
  • Relative(Induced) topology
  • Alternative methods of defining a topolgy in terms of neighbourhood system and Kuratowski closure operator
  • Comparison of topologies on a set,Intersection and union of topologies on a set

2

  • Continuous functions

Must know

15

  • Open and closed functions
  • Homeomorphism
  • Connected and its characterization
  • Connected subsets and their properties
  • Continuity and connectedness
  • Components
  • Locally connected spaces

3

  • Compact spaces and subsets

Must know

15

  • Compactness in terms of finite intersection property
  • Continuity and compact sets
  • Basic properties of compactness
  • Closedness of compactsubset and a continuous map from a compact space into a Hausdorff and its consequence
  • Sequentially and countably compact sets
  • Local compactness and one point compatification

4

  • First countable, Second countable and separable spaces

Nice to know

12

  • Hereditary and topological property
  • Countability of a collection of disjoint open sets in separable and second countable spaces
  • Lindelof theorem
  • T0,T1,T2(Hausdorff) separation axioms, their characterization and basic properties

Books Recommended:

  •    George F. Simmons, Introduction to Topology and Modern Analysis, McGraw- Hill Book Company, 1963.
  •    K.D. Joshi, Introduction to General Topology, Wiley Eastern Ltd.
  •    J.L. Kelly, General  Topology , Affiliated East West Press Pvt. Ltd., New Delhi.
  •    J.R. Munkres, Topology, Pearson Education Asia, 2002.
  •    W.J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 1964.
Admission 2017