Monotonically normal space

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 $A\subset A^{\prime }$ and $B^{\prime }\subset B$ with all four sets being closed, and $B$ disjoint from $B^{\prime }$, we have:

$G(A,B)\subset G(A^{\prime },B^{\prime })$

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

Relation with other properties

Stronger properties

• Metrizable space
• Ordered field-metrizable space
• Linearly orderable space
• Elastic space

Weaker properties

• Hereditarily collectionwise normal space
• Hereditarily normal space
• Collectionwise normal space
• Normal space

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