Connected implies no proper open subgroup

From Groupprops
Revision as of 08:12, 7 September 2007 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Statement

A connected topological group has no proper open subgroup.

Proof

Since every open subgroup is closed, a proper open subgroup is a nonempty subset that is both open and closed. The existence of such a subset contradicts connectedness.