Topological group

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Definition

A topological group (AKA: continuous group) is a $3$-tuple, $(G,*,\mathcal{J})$ where $G$ is a set, $*:G\times G\rightarrow G$ is a binary operation (a map where we write $a*b$ rather than $*(a,b)$) such that $(G,*)$ is a group and a topology, $\mathcal{J}$ on $G$ such that $(G,\mathcal{J})$ is a topological space, with the following two properties:

1. $m:G\times G\rightarrow G$ with $m:(x,y)\mapsto x*y$ is continuous (where $G\times G$ is considered with the product topology.
2. $i:G\rightarrow G$ with $i:x\rightarrow x^{-1}$ is also continuous
   • where $x^{-1}$ denotes the inverse element of $x$, $-x$ should be used if the group is denoted additively (see group page for more information)

Terminology

Given a topological group, $(G,*,\mathcal{J})$ we call the parts the following:

• Underlying set: $G$.
• Underlying group: $(G,*)$
• Underlying (topological) space: $(G,\mathcal{J})$

Examples

1. $(\mathbb{R}^2-\{0\},*)$