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.
A topological space
is called monotonically normal if it satisfies any of the following equivalent definitions:[1][2][3][4] The space
that assigns to each ordered pair
of disjoint closed sets in
an open set
is a normal space, as witnessed by the function
Condition (ii) says that
varies in a monotone fashion, hence the terminology monotonically normal.
is called a monotone normality operator.
to satisfy the property by replacing each
that assigns to each ordered pair
of separated sets in
) an open set
satisfying the same conditions (i) and (ii) of Definition 1.
μ
that assigns to each pair
an open set
μ ( x ,
automatically satisfies (Reason: Suppose
is T1, there is an open neighborhood
By condition (ii),
be a base for the topology of
that assigns to each pair
an open set
satisfying the same conditions (i) and (ii) of Definition 3.
that assigns to each pair
an open set
automatically satisfies all conditions of Definition 3.