Topological group

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Definition

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

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

Terminology

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

• Underlying set: [ilmath]G[/ilmath].
• Underlying group: [ilmath](G,*)[/ilmath]
• Underlying (topological) space: [ilmath](G,\mathcal{J})[/ilmath]

Examples

1. [ilmath](\mathbb{R}^2-\{0\},*)[/ilmath]