Metric outer measure

In mathematics, a monotonically normal space is a particular kind of normal space, with some special characteristics, and is such that it is hereditarily normal, and any two separated subsets are strongly separated. They are defined in terms of a monotone normality operator.

A $T_1$ topological space $(X,\mathcal{𝒯})$ is said to be monotonically normal if the following condition holds:

For every $x\in G$, where G is open, there is an open set $\mu (x,G)$ such that

1. $x\in \mu (x,G)\subseteq G$
2. if $\mu (x,G)\cap \mu (y,H)\ne \mathrm{\varnothing }$ then either $x\in H$ or $y\in G$.

There are some equivalent criteria of monotone normality.

Equivalent definitions

Definition 2

A space X is called monotonically normal if it is $T_1$ and for each pair of disjoint closed subsets $A,B$ there is an open set $G(A,B)$ with the properties

1. $A\subseteq G(A,B)\subseteq G(A,B{)}^{-}\subseteq X\mathrm{\setminus }B$ and
2. $G(A,B)\subseteq G(A^{\prime },B^{\prime })$, whenever $A\subseteq A^{\prime }$ and $B^{\prime }\subseteq B$.

This operator $G$ is called monotone normality operator.

Note that if G is a monotone normality operator, then $\tilde{G}$ defined by $\tilde{G}(A,B)=G(A,B)\mathrm{\setminus }G(B,A{)}^{-}$ is also a monotone normality operator; and $\tilde{G}$ satisfies

$\begin{array}{r}\tilde{G}(A,B)\cap \tilde{G}(B,A)=\mathrm{\varnothing }\end{array}$

For this reason we some time take the monotone normality operator so as to satisfy the above requirement; and that facilitates the proof of some theorems and of the equivalence of the definitions as well.

Definition 3

A space X is called monotonically normal if it is $T_1$,and to each pair (A, B) of subsets of X, with $A\cap B^{-}=\mathrm{\varnothing }=B\cap A^{-}$, one can assign an open subset G(A, B) of X such that

1. $A\subseteq G(A,B)\subseteq G(A,B{)}^{-}\subseteq X\mathrm{\setminus }B,$
2. $G(A,B)\subseteq G(A^{\prime },B^{\prime })\text{ whenever }A\subseteq A^{\prime }\text{ and }{B}^{\prime }\subseteq B$.

Definition 4

A space X is called monotonically normal if it is $T_1$ and there is a function H that assigns to each ordered pair (p,C) where C is closed and p is without C, an open set H(p,C) satisfying:

1. $p\in H(p,C)\subseteq X\mathrm{\setminus }C$
2. if D is closed and $p\in ̸C\supseteq D$ then $H(p,C)\subseteq H(p,D)$
3. if $p\ne q$ are points in X, then $H(p,\{q\})\cap H(q,\{p\})=\mathrm{\varnothing }$.

Properties

An important example of these spaces would be, assuming Axiom of Choice, the linearly ordered spaces; however, it really needs axiom of choice for an arbitrary linear order to be normal (see van Douwen's paper). Any generalised metric is monotonically normal even without choice. An important property of monotonically normal spaces is that any two separated subsets are strongly separated there. Monotone normality is hereditary property and a monotonically normal space is always normal by the first condition of the second equivalent definition.

We list up some of the properties :

1. A closed map preserves monotone normality.
2. A monotonically normal space is hereditarily collectionwise normal.
3. Elastic spaces are monotonically normal.

Some discussion links

1. R. W. Heath; D. J. Lutzer; P. L. Zenor, Monotonically Normal Spaces, Transactions of the American Mathematical Society, Vol. 178. (Apr., 1973), pp. 481-493.
2. Carlos R. Borges, A study of monotonically normal spaces, Proceedings of the American Mathematical Society, Vol. 38, No. 1. (Mar., 1973), pp. 211-214.
3. Eric K. van Douwen, Horrors of Topology Without AC: A Nonnormal Orderable Space, Proceedings of the American Mathematical Society, Vol. 95, No. 1. (Sep., 1985), pp. 101-105.
4. P. M. Gartside, Cardinal Invariants of Monotonically Normal Spaces, can be found here in Topology Atlas.
5. Henno Brandsma's discussion about Monotone Normality in Topology Atlas can be viewed here