ABOUT THE COURSE:Point set topology is one of the most important and basic courses that one encounters during a masters program in mathematics. This course introduces students to the most important concepts in point set topology. We begin the course by defining topological spaces and introducing various ways to put topologies on sets. Then we introduce the notion of continuous maps which enables us to see how different topological spaces interact with each other. A very special class of topological spaces is metric spaces. Most of our intuition for topology comes from metric spaces. We introduce metric spaces and try to understand the concepts we have learnt so far in this special case. After this we study the topological properties of connectedness, compactness and local compactness. We then introduce another method to put a topology on a set, namely the quotient topology. Finally we end the course with a discussion on when a topology arises from a metric. The main result in this part is Urysohn's Metrization Theorem.Throughout this course, the emphasis will be on various examples which we encounter often in mathematics, like Euclidean spaces, spheres, subsets of Euclidean spaces, matrices, general linear group, special linear group, orthogonal group, unitary group, special orthogonal group, special unitary group, Grassmannians, projective spaces. Our aim will be to try and see if these spaces are connected, path connected, Hausdorff, compact, locally compact. The focus of this course is not going to be on constructing counterexamples. There are far too many useful things in topology to focus on, and so I feel it would be better to learn these, than spend time on counterexamples which serve little purpose later.This course will be useful and accessible to students doing a masters in mathematics. The course can also be useful to students in a bachelors program in mathematics, physics or engineering streams, who need mathematics in their studies or have some interest in pure mathematics.INTENDED AUDIENCE:Masters Level Students in Mathematics. However, students in a bachelors program in mathematics or physics or engineering streams, who have had some exposure to set theory and real analysis should also be able to follow easily.PREREQUISITES:Students should know the basics of set theory. Some exposure to real analysis may be useful. Please see Assignment 0.
Overview
Syllabus
Week 1: Definition and examples of topological spaces, Examples of topological spaces, Basis for topology, Subspace Topology, Product Topology.Week 2: Continuous maps, Continuity of addition and multiplication maps, ring of continuous functions, Continuous maps to a product, Projection from a point.
Week 3: Closed subsets, Closure, Joining continuous maps, Metric spaces, Connectedness.Week 4:Connected components, Path connectedness.Week 5: Connectedness of GL(n,R)^+, Connectedness of GL(n,C), SL(n,C), SL(n,R), Hausdorff topological spaces, Compactness.
Week 6: SO(n) is connected, Compact metric spaces, Lebesgue Number Lemma, Locally compact spaces.Week 7:One point compactification, One point compactification (continued), Uniqueness of one point compatification, Quotient topology, Quotient topology on G/H.Week 8: Grassmannian, Normal topological spaces, Urysohn’s Lemma, Tietze Extension Theorem, Regular and Second Countable spaces, Urysohn’s Metrization Theorem.
Week 3: Closed subsets, Closure, Joining continuous maps, Metric spaces, Connectedness.Week 4:Connected components, Path connectedness.Week 5: Connectedness of GL(n,R)^+, Connectedness of GL(n,C), SL(n,C), SL(n,R), Hausdorff topological spaces, Compactness.
Week 6: SO(n) is connected, Compact metric spaces, Lebesgue Number Lemma, Locally compact spaces.Week 7:One point compactification, One point compactification (continued), Uniqueness of one point compatification, Quotient topology, Quotient topology on G/H.Week 8: Grassmannian, Normal topological spaces, Urysohn’s Lemma, Tietze Extension Theorem, Regular and Second Countable spaces, Urysohn’s Metrization Theorem.
Taught by
Prof. Ronnie Sebastian
