Monotonically normal space

In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal.

Definition
A topological space $$X$$ is called monotonically normal if it satisfies any of the following equivalent definitions:

Definition 1
The space $$X$$ is T1 and there is a function $$G$$ that assigns to each ordered pair $$(A,B)$$ of disjoint closed sets in $$X$$ an open set $$G(A,B)$$ such that:
 * (i) $$A\subseteq G(A,B)\subseteq \overline{G(A,B)}\subseteq X\setminus B$$;
 * (ii) $$G(A,B)\subseteq G(A',B')$$ whenever $$A\subseteq A'$$ and $$B'\subseteq B$$.

Condition (i) says $$X$$ is a normal space, as witnessed by the function $$G$$. Condition (ii) says that $$G(A,B)$$ varies in a monotone fashion, hence the terminology monotonically normal. The operator $$G$$ is called a monotone normality operator.

One can always choose $$G$$ to satisfy the property
 * $$G(A,B)\cap G(B,A)=\emptyset$$,

by replacing each $$G(A,B)$$ by $$G(A,B)\setminus\overline{G(B,A)}$$.

Definition 2
The space $$X$$ is T1 and there is a function $$G$$ that assigns to each ordered pair $$(A,B)$$ of separated sets in $$X$$ (that is, such that $$A\cap\overline{B}=B\cap\overline{A}=\emptyset$$) an open set $$G(A,B)$$ satisfying the same conditions (i) and (ii) of Definition 1.

Definition 3
The space $$X$$ is T1 and there is a function $$\mu$$ that assigns to each pair $$(x,U)$$ with $$U$$ open in $$X$$ and $$x\in U$$ an open set $$\mu(x,U)$$ such that:
 * (i) $$x\in\mu(x,U)$$;
 * (ii) if $$\mu(x,U)\cap\mu(y,V)\ne\emptyset$$, then $$x\in V$$ or $$y\in U$$.

Such a function $$\mu$$ automatically satisfies
 * $$x\in\mu(x,U)\subseteq\overline{\mu(x,U)}\subseteq U$$.

(Reason: Suppose $$y\in X\setminus U$$. Since $$X$$ is T1, there is an open neighborhood $$V$$ of $$y$$ such that $$x\notin V$$. By condition (ii), $$\mu(x,U)\cap\mu(y,V)=\emptyset$$, that is, $$\mu(y,V)$$ is a neighborhood of $$y$$ disjoint from $$\mu(x,U)$$.  So $$y\notin\overline{\mu(x,U)}$$.)

Definition 4
Let $$\mathcal{B}$$ be a base for the topology of $$X$$. The space $$X$$ is T1 and there is a function $$\mu$$ that assigns to each pair $$(x,U)$$ with $$U\in\mathcal{B}$$ and $$x\in U$$ an open set $$\mu(x,U)$$ satisfying the same conditions (i) and (ii) of Definition 3.

Definition 5
The space $$X$$ is T1 and there is a function $$\mu$$ that assigns to each pair $$(x,U)$$ with $$U$$ open in $$X$$ and $$x\in U$$ an open set $$\mu(x,U)$$ such that:
 * (i) $$x\in\mu(x,U)$$;
 * (ii) if $$U$$ and $$V$$ are open and $$x\in U\subseteq V$$, then $$\mu(x,U)\subseteq\mu(x,V)$$;
 * (iii) if $$x$$ and $$y$$ are distinct points, then $$\mu(x,X\setminus\{y\})\cap\mu(y,X\setminus\{x\})=\emptyset$$.

Such a function $$\mu$$ automatically satisfies all conditions of Definition 3.

Examples

 * Every metrizable space is monotonically normal.
 * Every linearly ordered topological space (LOTS) is monotonically normal. This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.
 * The Sorgenfrey line is monotonically normal. This follows from Definition 4 by taking as a base for the topology all intervals of the form $$[a,b)$$ and for $$x\in[a,b)$$ by letting $$\mu(x,[a,b))=[x,b)$$.  Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space.
 * Any generalised metric is monotonically normal.

Properties

 * Monotone normality is a hereditary property: Every subspace of a monotonically normal space is monotonically normal.
 * Every monotonically normal space is completely normal Hausdorff (or T5).
 * Every monotonically normal space is hereditarily collectionwise normal.
 * The image of a monotonically normal space under a continuous closed map is monotonically normal.
 * A compact Hausdorff space $$X$$ is the continuous image of a compact linearly ordered space if and only if $$X$$ is monotonically normal.