Property of topological spaces stronger than normality
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
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 base for the 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.