Semiregular space: Difference between revisions

Latest revision as of 17:01, 28 January 2012

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A semiregular space is a topological space $X$ satisfying the following equivalent conditions:

The regular open subsets (these are subsets that equal the interior of their closure) form a basis for the space.
For any $x \in X$ and any open subset $V \subseteq X$ containing $x$ , there exists a regular open subset $U$ of $X$ containing $x$ and contained in $V$ .

Note that regular open is not the same as being open and regular in the subspace topology. For the notion defined using that, see locally regular space.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular space		regular implies semiregular	semiregular not implies regular

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 ==Definition==
-A '''semiregular space''' is a [[T1 space]] in which the [[regular open subset]]s (these are subsets that equal the [[interior]] of their [[closure]]) form a [[basis]] for the space.
+A '''semiregular space''' is a [[topological space]] <math>X</math>satisfying the following equivalent conditions:
+# The [[regular open subset]]s (these are subsets that equal the [[interior]] of their [[closure]]) form a [[basis]] for the space.
+# For any <math>x \in X</math> and any open subset <math>V \subseteq X</math> containing <math>x</math>, there exists a [[regular open subset]] <math>U</math> of <math>X</math> containing <math>x</math> and contained in <math>V</math>.
 Note that ''regular open'' is not the same as being open and regular in the [[subspace topology]]. For the notion defined using that, see [[locally regular space]].