Locally regular space: Difference between revisions

Latest revision as of 17:24, 28 January 2012

Definition

A topological space $X$ is termed locally regular if it satisfies the following equivalent conditions:

It has a basis of open subsets each of which is a regular space under the subspace topology.
For any $x \in X$ , there exists an open subset $U$ containing $x$ such that $U$ is a regular space with the subspace topology.
For any $x \in X$ and open subset $V$ containing $x$ , there exists an open subset $U$ containing $x$ such that $U \subseteq V$ and $U$ is a regular space with the subspace topology.

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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular space

@@ Line 4: / Line 4: @@
 # It has a [[basis]] of [[open subset]]s each of which is a [[regular space]] under the [[subspace topology]].
+# For any <math>x \in X</math>, there exists an [[open subset]] <math>U</math> containing <math>x</math> such that <math>U</math> is a [[regular space]] with the [[subspace topology]].
 # For any <math>x \in X</math> and open subset <math>V</math> containing <math>x</math>, there exists an open subset <math>U</math> containing <math>x</math> such that <math>U \subseteq V</math> and <math>U</math> is a [[regular space]] with the [[subspace topology]].
+{{topospace property}}
 ==Relation with other properties==