Monotonically normal space

A space X is called monotonically normal if it is ${\displaystyle T_{1}}$ and for each pair of disjoint closed subsets ${\displaystyle A,B}$ there is an open set ${\displaystyle G(A,B)}$ with the properties

1. ${\displaystyle A\subseteq G(A,B)\subseteq G(A,B)^{-}\subseteq X\backslash B}$ and
2. ${\displaystyle G(A,B)\subseteq G(A',B')}$, whenever ${\displaystyle A\subseteq A'}$ and ${\displaystyle B'\subseteq B}$.

This operator ${\displaystyle G}$ is called monotone normality operator.

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

${\displaystyle {\begin{aligned}{\tilde {G}}(A,B)\cap {\tilde {G}}(B,A)=\emptyset \end{aligned}}}$

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.

A space X is called monotonically normal if it is ${\displaystyle T_{1}}$,and to each pair (A, B) of subsets of X, with ${\displaystyle A\cap B^{-}=\emptyset =B\cap A^{-}}$, one can assign an open subset G(A, B) of X such that

1. ${\displaystyle A\subseteq G(A,B)\subseteq G(A,B)^{-}\subseteq X\backslash B,}$
2. ${\displaystyle G(A,B)\subseteq G(A',B'){\mbox{ whenever }}A\subseteq A'{\mbox{ and }}B'\subseteq B}$.

A space X is called monotonically normal if it is ${\displaystyle T_{1}}$ 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. ${\displaystyle p\in H(p,C)\subseteq X\backslash C}$
2. if D is closed and ${\displaystyle p\not \in C\supseteq D}$ then ${\displaystyle H(p,C)\subseteq H(p,D)}$
3. if ${\displaystyle p\neq q}$ are points in X, then ${\displaystyle H(p,\{q\})\cap H(q,\{p\})=\emptyset }$.