# subgroup of topological group is either clopen or has empty interior

Theorem - Every subgroup of a topological group^{} is either clopen or has empty interior.

$$

*Proof:* Let $G$ be a topological group and $H\subseteq G$ a subgroup. Suppose the interior of $H$ is nonempty, i.e. there is a non-empty open set $U$ of $G$ such that $U\subseteq H$. Translating $U$ around $H$ we can see that $H$ is open: if $u\in U$ then for every $h\in H$ the set $h{u}^{-1}U$ is open in $G$, is contained in $H$ and contains $h$, which implies that $H$ is open in $G$.

Let us now see that $H$ is closed. Let $\overline{H}$ denote the closure of $H$ and let ${H}^{2}$ be the set of elements of the form ${h}_{1}{h}_{2}$ where ${h}_{1},{h}_{2}\in H$. Of course, since $H$ is a subgroup of $G$, we have that ${H}^{2}=H$. Also, since $H$ is open we know that$H\subseteq \overline{H}\subseteq {H}^{2}$ (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups) - 5). Hence $\overline{H}=H$, i.e. $H$ is closed.

We have proven that a subgroup of a topological group must be clopen or it must have empty interior. Since this two topological properties can never be satisfied simultaneously, we have that every subgroup of a topological group is either clopen or it has empty interior. $\mathrm{\square}$

Title | subgroup of topological group is either clopen or has empty interior |
---|---|

Canonical name | SubgroupOfTopologicalGroupIsEitherClopenOrHasEmptyInterior |

Date of creation | 2013-03-22 18:01:29 |

Last modified on | 2013-03-22 18:01:29 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 6 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 22A05 |