Property of topological spaces stronger than normality
In mathematics, specifically in the field of
hereditarily normal
.
Definition
A topological space is called monotonically normal if it satisfies any of the following equivalent definitions:[1][2][3][4]
Definition 1
The space is T1 and there is a function that assigns to each ordered pair of disjoint closed sets in an open set such that:
(i) ;
(ii) whenever and .
Condition (i) says is a normal space, as witnessed by the function .
Condition (ii) says that varies in a monotone fashion, hence the terminology monotonically normal.
The operator is called a monotone normality operator.
One can always choose to satisfy the property
,
by replacing each by .
Definition 2
The space is T1 and there is a function that assigns to each ordered pair of separated sets in (that is, such that ) an open set satisfying the same conditions (i) and (ii) of Definition 1.
Definition 3
The space is T1 and there is a function that assigns to each pair with open in and an open set such that:
(i) ;
(ii) if , then or .
Such a function automatically satisfies
.
(Reason: Suppose . Since is T1, there is an open neighborhood of such that . By condition (ii), , that is, is a neighborhood of disjoint from . So .)[5]
Definition 4
Let be a
topology
of .
The space is T1 and there is a function that assigns to each pair with and an open set satisfying the same conditions (i) and (ii) of Definition 3.
Definition 5
The space is T1 and there is a function that assigns to each pair with open in and an open set such that:
(i) ;
(ii) if and are open and , then ;
(iii) if and are distinct points, then .
Such a function automatically satisfies all conditions of Definition 3.
This follows from Definition 4 by taking as a base for the topology all intervals of the form and for by letting . Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the