Topological Groups Notes Patrick Da Silva February 19, 2014 This document is describing and proving a collection of propositions concerning topological spaces and topological groups. In particular, it shows that if G is a topological group, then letting H = {1}, we show that G/H is a T 3 1 2 topological group. Definition 1. (Topological basics) A topological space is a set X together with a topology T , where T ⊆P (X ) def = {A | A ⊆ X }. The topological space is denoted by (X, T ) or simply by X if the topology is understood from the context. The topology must be such that - ∅ ∈T , X ∈T - If A i ∈T for any i ∈ I where I is some set, then  i∈I A i ∈T - If A i ∈T for i =1,...,n, then  n i=1 A i ∈T . The elements of T are called the open subsets of X and their complements are called the closed subsets of X , that is, A ⊆ X is closed if and only if X \A ∈T . For x ∈ X , a subset A ⊆ X is said to be a neighborhood of x if x ∈ A and A is open. The closure of a subset A ⊆ X is the intersection of all closed subsets of X containing A. It is denoted by A. The interior of a subset A ⊆ X is the union of all open subsets of X contained in A. It is denoted by int(A). 1