Monotonically normal space

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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' and B \subset B' with all four sets being closed, and B disjoint from B', we have:

G(A,B) \subset 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

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.