MT3134- Point set topology (LHC 108 in IISER Pune)
Reference: James Munkres, Topology I
Office hours: (Office number: A-459) Friday 11:00 am to 12:00 pm
Lecture 0 - (Amit Hogadi) Introduction to the course.
Lecture 1 (22/08/2022)- Definition of topology on a set, Examples of discrete, trivial, finite complement topologies, metric spaces, fineness/coarseness of topologies on a set, Definition of a basis for a topology and examples. The definition of the topology generated by a basis and the proof that it is indeed a topology.
Lecture 2 (23/08/2022)- Open subsets are unions of basis elements (Lemma 13.1), criterion (Lemma 13.2) for a collection of open subsets to be a basis for the topology, examples of various bases on usual plane.
Lecture 3 (25/08/2022)- Criterion (Lemma 13.3) for comparison of two topologies on a set in terms of their bases, various topologies on the real line R: standard topology, lower limit topology, K-topology; and their comparison to each other.
Notes for Lectures 1-2-3 are here.
Lecture 4 (29/08/2022)- Product topology, examples, basis for product topology in terms of basis for topologies of factor, subspace topology, Product topology on product of subspaces is the same as the subspace topology on the cartesian product of subsets of the cartesian product of spaces. notes
Lecture 5 (30/08/2022)- Some problems from assignment 1 were discussed, Closed sets, closed sets in subspace topology notes
Lecture 6 (1/09/2022)- Interior of a set, closure of a set, limit point of a set, Criterion of a point in the space to be in a closure of a subset, Closure of a set in terms of its limit points, Discussion on Hausdorff condition started and to be continued in the next talk. notes
Lecture 7 (5/09/2022)- Side-discussion on dictionary order on R^2, Spaces satisfying T_1-axiom, Hausdorff topological spaces, Definition of a continuous function between topological spaces. notes
Lecture 8 (6/09/2022)- Equivalent criteria for continuity of functions between topological spaces, Homeomorphisms, Examples, Imbeddings and examples notes
Lecture 9 (8/09/2022)- New continuous functions out of the known ones, pasting lemma for continuous maps, Maps into products notes
Lecture 10 (12/09/2022)- Metric topology, Euclidean metric and square metrics on R^n, criterion for comparing two metric topologies, Product topology, topologies induced by Euclidean metric and square metric on R^n all agree with each other. notes
Lecture 11 (13/09/2022)- Further properties of metric spaces, sequence lemma, continuity in terms of the metric, uniform convergence of a sequence of continuous functions. notes
Lecture 12 (15/09/2022)- Quotient map, quotient topology, Quotient spaces, Examples of construction of sphere from unit disc using the quotient spaces. notes
Lecture 13 (19/09/2022)- Examples of sphere and torus as quotient spaces, Continuous maps from quotient spaces, Example of toplogical groups and quotient spaces notes
Lecture 14 (20/09/2022)- Connectedness, results about connected topological spaces notes
Lecture 15 (22/09/2022)- Connected subsets of real line, Intermediate value theorem in topology, Path connectedness and examples notes
Assignments
Assignment 1 (based on material covered in week 1)
Assignment 2 (based on material covered in week 2)
Assignment 3 (based on material covered in week 3)
Assignment 4 (based on material covered in week 4)
Assignment 5 (based on material covered in week 5)