Monotonically normal space: Difference between revisions

Revision as of 19:26, 24 October 2009

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

This is a variation of normality. View other variations of normality

This article or section of article is sourced from:Wikipedia

Definition

Definition with symbols

A topological space $X$ is termed monotonically normal if there exists an operator $G$ from ordered pairs of disjoint closed sets to open sets, such that:

For any disjoint closed subsets $A,B$ , $G(A,B)$ contains $A$ and its closure is disjoint from $B$
If $A\subseteq A'$ and $B'\subseteq B$ with all four sets being closed, and $B$ disjoint from $A'$ , we have:

$G(A,B)\subseteq G(A',B')$

This is the monotonicity condition. Such an operator $G$ is termed a monotone normality operator.

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties

Perfectly normal space

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a monotonically normal space is monotonically normal. For full proof, refer: Monotone normality is hereditary

@@ Line 11: / Line 11: @@
 A [[topological space]] <math>X</math> is termed '''monotonically normal''' if there exists an operator <math>G</math> from ordered pairs of disjoint closed sets to open sets, such that:
-* For any disjoint closed subsets <math>A,B</math>, <math>G(A,B)</math> contains <math>A</math> and its closure is disjoint from <math>B</math>
+# For any disjoint closed subsets <math>A,B</math>, <math>G(A,B)</math> contains <math>A</math> and its closure is disjoint from <math>B</math>
-* If <math>A \subset A'</math> and <math>B' \subset B</math> with all four sets being closed, and <math>B</math> disjoint from <math>B'</math>, we have:
+# If <math>A \subseteq A'</math> and <math>B' \subseteq B</math> with all four sets being closed, and <math>B</math> disjoint from <math>A'</math>, we have:
-<math>G(A,B) \subset G(A',B')</math>
+<math>G(A,B) \subseteq G(A',B')</math>
 This is the ''monotonicity'' condition. Such an operator <math>G</math> is termed a monotone normality operator.