Essentials of Topology

ABOUT THE COURSE: The objective of this course is to explore the foundations of mathematics at a level and at depth for someone ambitious to study higher mathematics. After completing the course, a student can realize what it means to do mathematics instead of only learning it or doing some computational exercises.INTENDED AUDIENCE: Under graduatePREREQUISITES: One course in real analysis available on NPTEL Portal is desirable.INDUSTRY SUPPORT: Any industry who practices Topological Data Analysis

Syllabus

Week 1: Topological spaces: Prerequisites, Open sets and topology, Examples, Comparison of topologies.Week 2:Topological spaces: Bases and Subbases for a topology, Neighborhoods, Closed sets.Week 3:Interior, Closure and Boundary: Interior and Closure of sets, Limit points, The boundary of a set, Dense sets.Week 4:Creating new topological spaces: The subspace topology, The product topology, The quotient topology.Week 5:Alternative methods of defining a topology in terms of Kuratowski closure/interior operator, First and Second countable spaces, Separable spaces.Week 6:Continuous functions and homeomorphisms, Non- homeomorphic spaces.Week 7:Connectedness: A first approach to connectedness, Distinguishing topological spaces via connectedness, Connected subspaces of the real line.Week 8:Connectedness: Components, Path connectedness, Local connectedness.Week 9:Compactness: Open covering and compact spaces, Basic properties of compactness. Compactness and finite intersection property.Week 10:Compactness: B-W compactness, Limit point compactness, Local compactness, One-point compactificationWeek 11:The separation axioms T0, T1, T2, T3, T4; their characterizations and basic properties. Urysohn’s lemma, Tietze extension theorem.Week 12:The Tychnoff Theorem, The Stone-Cech Compactification