Introduction to Topology

NA

Objectives

NA

Prerequisites

Real Analysis

Syllabus

Topological spaces (definitions of topological space and topology. Main topological notions, topology and neighborhood bases, topological subspaces). Continuous functions in topological spaces (pointwise (local) continuity and total continuity, properties of continuous functions, the product topology, metric topologies). Convergence (nets and subnets, convergence of sequences, convergence of nets, study of a function's continuity by using nets). Compactness (definition of a compact topological space and basic properties, continuity of functions and compactness, compact metric spaces). Connection (definition of a connected topological space and basic properties, connected components, continuity of functions and connection). Countability and separation axioms : Urysohn's and Tychonoff's theorems (Urysohn's Lemma, Urysohn's metrization theorem, Tychonoff's Theorem). Topologies of function spaces (the topology of pointwise convergence, the compact-open topology).