Topology
Teacher
Dr. Swarnalata Jena
Category
Core Courses
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Course Name : Topology
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
Course 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.
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 |
Understanding elementary properties of topological spaces and structures defined on them
|
PO1(3), PO9(2) |
| CO2 |
Construct maps between topological spaces ability to handle abstract ideas of Mathematics and Mathematical proofs
|
PO4(3), PO2(2), |
| CO4 |
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.
|
PO3(2) |
| CO5 |
Prove basic results about completeness, compactness, connectedness and convergence within these structures.
|
PO5(2), PO9(2) |
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.
2. http://mat.uab.cat/ret/sites/default/files/material/otras_contribuciones/ProceedingsWIAT10.pdf
Session Plan
Session 1
Introduction of topological space
https://www.youtube.com/watch?v=AmgkSdhK4K8
https://www.slideshare.net/MubinaAppiPune/topology-77688404
Session 2
Open sets and limit points
https://www.youtube.com/watch?v=q6698vWK3ts
https://www.slideshare.net/ramyaramya47/topology-definitions
Session 3
Session 4
Bases and relative topologies
http://www.math.toronto.edu/ivan/mat327/docs/notes/02-bases.pdf
https://www.youtube.com/watch?v=yZoW1CUBpvo
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 10
Session 14 & 15
Project 3 (2 hrs)
Sober topological space
https://ncatlab.org/nlab/show/sober+topological+space
Session 19
Session 20
Session 23
Completely regular spaces
https://www.youtube.com/watch?v=QyYhvu5FQno
https://mathworld.wolfram.com/CompletelyRegularSpace.html
Session 26
Urysohn’s metrization theorem
https://www.youtube.com/watch?v=AYDAR3I8Jr8
http://www.math.toronto.edu/ivan/mat327/docs/notes/15-umt.pdf
Session 30
Session 31 & 32
Project 5 (2 hrs)
Scott topology
https://www.youtube.com/watch?v=A7_lEw_Ene8&t=68s
https://ncatlab.org/nlab/show/Scott+topology
Session 33 & 34
Project 6 (2 hrs)
Scott Continuity
https://we.vub.ac.be/sites/default/files/files/bachelor_scriptie_filip_moons2.pdf
https://www.youtube.com/watch?v=A7_lEw_Ene8&t=83s
Session 35
Session 37
Session 39
Session 41
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