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Curriculum | Elementary Topology (Code-09050604)

Curriculum

Elementary Topology  (Code-09050604)

S.No.

Content of the topics

Learning Objectives

Teaching Guidelines

Methodology

Time                         (Hours)

1

  • Statements only of (Axiom of choice, Zorn’s Lemma, Well ordering theorem and Continuum hypothesis)
Students will be able to study of basics of topology

Lecture should be effective so that student will be able to grasp the topics easily

Assignments/ Seminars/    Class Tests/ Presentations

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
Student will be able to study of continuous functions and connectedness

Lecture should be effective so that student will be able to grasp the topics easily

Assignments/ Seminars/    Class Tests/ Presentations

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
Student will be able to Study the compactness

Lecture should be effective so that student will be able to grasp the topics easily

Assignments/ Seminars/    Class Tests/ Presentations

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
Student will be able to know about the countable spaces

Lecture should be effective so that student will be able to grasp the topics easily

Assignments/ Seminars/    Class Tests/ Presentations

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