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单词 SubgroupOfTopologicalGroupIsEitherClopenOrHasEmptyInterior
释义

subgroup of topological group is either clopen or has empty interior


Theorem - Every subgroup of a topological groupMathworldPlanetmath is either clopen or has empty interior.

Proof: Let G be a topological group and HG a subgroup. Suppose the interior of H is nonempty, i.e. there is a non-empty open set U of G such that UH. Translating U around H we can see that H is open: if uU then for every hH the set hu-1U 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 H¯ denote the closure of H and let H2 be the set of elements of the form h1h2 where h1,h2H. Of course, since H is a subgroup of G, we have that H2=H. Also, since H is open we know thatHH¯H2 (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups) - 5). Hence 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.

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更新时间:2025/5/4 16:00:11