Marginal implies unconditionally closed: Difference between revisions
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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 [[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>. | ||
In particular, the result applies to the cases that <math>G</math> is a [[topological group]], [[Lie group]], or [[algebraic group]]. | |||
==Related facts== | ==Related facts== | ||
Revision as of 23:59, 14 January 2012
Statement
Suppose is a semitopological group and is a marginal subgroup of as an abstract group. Then, is a closed subgroup of (i.e., it is a closed subset in the topological sense). In fact, is a closed normal subgroup of .
In particular, the result applies to the cases that is a topological group, Lie group, or algebraic group.