# compact spaces with group structure

Proposition^{}. Assume that $(G,M)$ is a group (with multiplication $M:G\times G\to G$) and $G$ is also a topological space^{}. If $G$ is compact^{} Hausdorff^{} and $M:G\times G\to G$ is continuous^{}, then $(G,M)$ is a topological group^{}.

Proof. Indeed, all we need to show is that function $f:G\to G$ given by $f(g)={g}^{-1}$ is continuous. Note, that the following holds for the graph of $f$:

$$\mathrm{\Gamma}(f)=\{(g,f(g))\in G\times G\}=\{(g,{g}^{-1})\in G\times G\}={M}^{-1}(e),$$ |

where $e$ denotes the neutral element in $G$. It follows (from continuity of $M$) that $\mathrm{\Gamma}(f)$ is closed in $G\times G$. It is well known (see the parent object for details) that this implies that $f$ is continuous, which completes^{} the proof. $\mathrm{\square}$

Title | compact spaces with group structure |
---|---|

Canonical name | CompactSpacesWithGroupStructure |

Date of creation | 2013-03-22 19:15:13 |

Last modified on | 2013-03-22 19:15:13 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 6 |

Author | joking (16130) |

Entry type | Corollary |

Classification | msc 26A15 |

Classification | msc 54C05 |