Marginal implies unconditionally closed: Difference between revisions

Revision as of 00:50, 17 July 2013

Suppose $G$ is a T0 quasitopological group (i.e., a quasitopological group whose underlying set is a T0 space) and $H$ is a marginal subgroup of $G$ as an abstract group. Then, $H$ is a closed subgroup of $G$ (i.e., it is a closed subset in the topological sense). In fact, $H$ is a closed normal subgroup of $G$ .

In particular, the result applies to the cases that $G$ is a T0 topological group, Lie group, or algebraic group.

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 ==Statement==
-Suppose <math>G</math> is a [[semitopological group]] and <math>H</math> is a [[fact about::marginal subgroup;1| ]][[marginal subgroup]] of <math>G</math> as an abstract group. Then, <math>H</math> is a [[closed subgroup]] of <math>G</math> (i.e., it is a closed subset in the topological sense). In fact, <math>H</math> is a [[closed normal subgroup]] of <math>G</math>.
+Suppose <math>G</math> is a [[T0 quasitopological group]] (i.e., a [[quasitopological group]] whose underlying set is a [[topospaces:T0 space|T0 space]]) and <math>H</math> is a [[fact about::marginal subgroup;1| ]][[marginal subgroup]] of <math>G</math> as an abstract group. Then, <math>H</math> is a [[closed subgroup]] of <math>G</math> (i.e., it is a closed subset in the topological sense). In fact, <math>H</math> is a [[closed normal subgroup]] of <math>G</math>.
-In particular, the result applies to the cases that <math>G</math> is a [[topological group]], [[Lie group]], or [[algebraic group]].
+In particular, the result applies to the cases that <math>G</math> is a [[T0 topological group]], [[Lie group]], or [[algebraic group]].
 ==Related facts==
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 ===Applications===
-* [[Center is closed in semitopological group]]
+* [[Center is closed in T0 quasitopological group]]