Monotonically normal space

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 formula_1 topological space formula_2 is said to be "monotonically normal" if the following condition holds:
For every formula_3, where G is open, there is an open set formula_4 such that
There are some equivalent criteria of monotone normality.
Equivalent definitions.
Definition 2.
A space X is called monotonically normal if it is formula_1 and for each pair of disjoint closed subsets formula_10 there is an open set formula_11 with the properties
This operator formula_16 is called monotone normality operator. 
Note that if G is a monotone normality operator, then formula_17 defined by formula_18 is also a monotone normality operator; and formula_17 satisfies 
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 formula_1, and to each pair (A, B) of subsets of X, with formula_22, one can assign an open subset G(A, B) of X such that
Definition 4.
A space X is called monotonically normal if it is formula_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:
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 :