# Topology

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# Code(Credit) : CUTM1536(3-0-1)

## Course Objectives

• To introduce the student to elementary properties of topological spaces and structures defined on them
• To introduce the student to maps between topological spaces
• To develop the student’s ability to handle abstract ideas of Mathematics and Mathematical proofs

## Learning Outcomes

• Understanding elementary properties of topological spaces and structures defined on them
• Construct maps between topological spaces
• Ability to handle abstract ideas of Mathematics and Mathematical proofs
• Demonstrate an understanding of the concepts of metric spaces and topological spaces, and their role in mathematics.
• Demonstrate familiarity with a range of examples of these structures.
• Prove basic results about completeness, compactness, connectedness and convergence within these structures.

## Course Syllabus

Module I
Introduction of topological space, Open sets and limit points, Closed sets and closure, Bases and relative topologies
Project 1: Applications of Topology to the Analysis of 1-Dimensional Objects
Project 2: Topologies sequentially equivalent to Kuratowski Painlev´e convergence

Module II
Connected sets and components,compact and Countable compact spaces , continuous functions, Homeomorphisms
Project 3: Sober topological space

Module III
To-and T1-spaces and sequence, Separation axioms

Module IV
Axioms of count ability, Regular and normal spaces, Completely regular spaces
Project 4: Upper Topology

Module V
Urysohn’s metrization theorem , Urysohn’s Lemma, Metrization, Tietze extension theorem
Project 5: Scott topology
Project 6: Scott continuity
Module VI

Finite products, product invariant properties, product topology

Module VII

Metric topology, Metric products, Dense set

Text Books
1. W. J. Pervin, Foundations of General Topology, Academic Press.
Chapters: 3 (3.1, 3.2 and 3.4), 4(4.1 to 4.4), 5 (5.1 to 5.3, 5.5 and 5.6), 8 (8.1 to 8.4), 10 (10.1 only).
2. J. R. Munkres; Topology – A First Course, Prentice Hall of India, 1996.
Reference Book
1. K. D. Joshi, Introduction to General Topology, Wiley Eastern Ltd., 1983.

## Session 1

Introduction of topological space

## Session 5

Bases and relative topologies theorems
https://www.youtube.com/watch?v=p2ofJPh2yMw

## Session 6 & 7

Project 1 (2 hrs)
Applications of Topology to the Analysis of 1-Dimensional Objects
https://www.youtube.com/watch?v=p2ofJPh2yMw
https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=17072

## Session 8 & 9

Project 2 (2 hrs)
Topologies sequentially equivalent to Kuratowski Painlev´e convergence

https://link.springer.com/chapter/10.1007/978-94-015-8149-3_5

## Session 11

compact and Countable compact spaces
https://www.youtube.com/watch?v=YmQnsqxMmlw

## Session 14 & 15

Project 3 (2 hrs)
Sober topological space
https://ncatlab.org/nlab/show/sober+topological+space

## Case Studies

### Mrs. Saubhagyalaxmi Singh

##### Assistant Professor in School of Applied Sciences, Department of Mathematics
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Mrs. Saubhagyalaxmi Singh works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. She has over 6 years of teaching experience with exposure to Utkal University, Bhubaneswar, Department of Mathematics. She has published 6 papers communicated to different International Journals. She has guided 7 M.Sc. projects in her area. She has […]