An invitation to Topology

Overview

Topology is the study of abstract shapes and their properties which do not change under stretching or squeezing an object without tearing. In this course, we shall investigate and formalize the abstract notion of a “space” under a bare minimum set-theoretic structure, called a topology. This basic structure gives rise to a beautiful and rich source of ideas which at the same time greatly simplify and extend familiar objects in mathematics, such as limits and continuous functions. Arguably, topology (along with the theory of numbers) lies at the heart of mathematics. I invite you to explore this abstract yet foundational theory which plays a significant role in modern mathematics, and by extension, other mathematical sciences.PRE-REQUISITES: NillINTENDED-AUDIENCE:NillINDUSTRY-SUPPORT: Nill

Syllabus

Week 1:Set theory and Logic: Sets and functions, Finite and Infinite sets, Well-ordered sets, Zorn’s lemma and the Axiom of ChoiceWeek 2:Topological spaces: the basic axioms of topology with examples, Bases and Subbases, Various kinds of topologies on the real lineWeek 3:Limit points and closed sets, continuous functionsWeek 4:Product topology, Quotient topology and Metric topologyWeek 5:Connectedness: connected spaces and subspaces, Connectedness of the real line, Intermediate value theoremWeek 6:Connected components, Path connected, locally connected, and locally path-connected spacesWeek 7:Compact spaces: open cover characterization, Finite intersection property, various notions of compactnessWeek 8:Compact subspaces of the real line, Heine-Borel theorem, extreme value theoremWeek 9:Urysohn’s Lemma and Tietze extension theorem on metric spacesWeek 10:Complete metric spaces, Totally bounded metric spaces and compactness, Lebesgue number lemmaWeek 11:Function spaces and the Arzela-Ascoli theoremWeek 12:Baire spaces and the Baire category theorem