Unconditionally closed subgroup: Difference between revisions

Latest revision as of 01:47, 28 July 2013

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

A subgroup $H$ of a group $G$ is termed an unconditionally closed subgroup if $H$ is a closed subgroup of $G$ for any topology on $G$ that turns $G$ into a T0 topological group.

Relation with other properties

Corresponding subset property

Unconditionally closed subset is the "subset" version of the property.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite subgroup				\|FULL LIST, MORE INFO
c-closed subgroup				\|FULL LIST, MORE INFO
algebraic subgroup				\|FULL LIST, MORE INFO
marginal subgroup				\|FULL LIST, MORE INFO
weakly marginal subgroup				\|FULL LIST, MORE INFO

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 | [[Weaker than::marginal subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|marginal subgroup}}
 |-
-| [[Weaker than::weakly marginal subgroup]] || || || || {{intermediaten notions short|unconditionally closed subgroup|weakly marginal subgroup}}
+| [[Weaker than::weakly marginal subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|weakly marginal subgroup}}
 |}