Monotonically normal space – Wikipedia

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[edit]

A topological space

X{displaystyle X}

is called monotonically normal if it satisfies any of the following equivalent definitions:^[1][2][3][4]

Definition 1[edit]

The space

X{displaystyle X}

is T₁ and there is a function

G{displaystyle G}

that assigns to each ordered pair

(A,B){displaystyle (A,B)}

of disjoint closed sets in

X{displaystyle X}

an open set

G(A,B){displaystyle G(A,B)}

such that:

(i) A⊆G(A,B)⊆G(A,B)¯⊆X∖B{displaystyle Asubseteq G(A,B)subseteq {overline {G(A,B)}}subseteq Xsetminus B}

;

(ii) G(A,B)⊆G(A′,B′){displaystyle G(A,B)subseteq G(A’,B’)}

whenever A⊆A′{displaystyle Asubseteq A’}

and B′⊆B{displaystyle B’subseteq B}

.

Condition (i) says

X{displaystyle X}

is a normal space, as witnessed by the function

G{displaystyle G}

.
Condition (ii) says that

G(A,B){displaystyle G(A,B)}

varies in a monotone fashion, hence the terminology monotonically normal.
The operator

G{displaystyle G}

is called a monotone normality operator.

One can always choose

G{displaystyle G}

to satisfy the property

G(A,B)∩G(B,A)=∅{displaystyle G(A,B)cap G(B,A)=emptyset }

,

by replacing each

G(A,B){displaystyle G(A,B)}

by

G(A,B)∖G(B,A)¯{displaystyle G(A,B)setminus {overline {G(B,A)}}}

.

Definition 2[edit]

The space

X{displaystyle X}

is T₁ and there is a function

G{displaystyle G}

that assigns to each ordered pair

(A,B){displaystyle (A,B)}

of separated sets in

X{displaystyle X}

(that is, such that

A∩B¯=B∩A¯=∅{displaystyle Acap {overline {B}}=Bcap {overline {A}}=emptyset }

) an open set

G(A,B){displaystyle G(A,B)}

satisfying the same conditions (i) and (ii) of Definition 1.

Definition 3[edit]

The space

X{displaystyle X}

is T₁ and there is a function

μ{displaystyle mu }

that assigns to each pair

(x,U){displaystyle (x,U)}

with

U{displaystyle U}

open in

X{displaystyle X}

and

x∈U{displaystyle xin U}

an open set

μ(x,U){displaystyle mu (x,U)}

such that:

(i) x∈μ(x,U){displaystyle xin mu (x,U)}

;

(ii) if μ(x,U)∩μ(y,V)≠∅{displaystyle mu (x,U)cap mu (y,V)neq emptyset }

, then x∈V{displaystyle xin V}

or y∈U{displaystyle yin U}

.

Such a function

μ{displaystyle mu }

automatically satisfies

x∈μ(x,U)⊆μ(x,U)¯⊆U{displaystyle xin mu (x,U)subseteq {overline {mu (x,U)}}subseteq U}

.

(Reason: Suppose

y∈X∖U{displaystyle yin Xsetminus U}

. Since

X{displaystyle X}

is T₁, there is an open neighborhood

V{displaystyle V}

of

y{displaystyle y}

such that

x∉V{displaystyle xnotin V}

. By condition (ii),

μ(x,U)∩μ(y,V)=∅{displaystyle mu (x,U)cap mu (y,V)=emptyset }

, that is,

μ(y,V){displaystyle mu (y,V)}

is a neighborhood of

y{displaystyle y}

disjoint from

μ(x,U){displaystyle mu (x,U)}

. So

y∉μ(x,U)¯{displaystyle ynotin {overline {mu (x,U)}}}

.)^[5]

Definition 4[edit]

Let

B{displaystyle {mathcal {B}}}

be a base for the topology of

X{displaystyle X}

.
The space

X{displaystyle X}

is T₁ and there is a function

μ{displaystyle mu }

that assigns to each pair

(x,U){displaystyle (x,U)}

with

U∈B{displaystyle Uin {mathcal {B}}}

and

x∈U{displaystyle xin U}

an open set

μ(x,U){displaystyle mu (x,U)}

satisfying the same conditions (i) and (ii) of Definition 3.

Definition 5[edit]

The space

X{displaystyle X}

is T₁ and there is a function

μ{displaystyle mu }

that assigns to each pair

(x,U){displaystyle (x,U)}

with

U{displaystyle U}

open in

X{displaystyle X}

and

x∈U{displaystyle xin U}

an open set

μ(x,U){displaystyle mu (x,U)}

such that:

(i) x∈μ(x,U){displaystyle xin mu (x,U)}

;

(ii) if U{displaystyle U}

and V{displaystyle V}

are open and x∈U⊆V{displaystyle xin Usubseteq V}

, then μ(x,U)⊆μ(x,V){displaystyle mu (x,U)subseteq mu (x,V)}

;

(iii) if x{displaystyle x}

and y{displaystyle y}

are distinct points, then μ(x,X∖{y})∩μ(y,X∖{x})=∅{displaystyle mu (x,Xsetminus {y})cap mu (y,Xsetminus {x})=emptyset }

.

Such a function

μ{displaystyle mu }

automatically satisfies all conditions of Definition 3.

Examples[edit]

• Every metrizable space is monotonically normal.^[4]
• Every linearly ordered topological space (LOTS) is monotonically normal.^[6][4] This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.^[7]
• The Sorgenfrey line is monotonically normal.^[4] This follows from Definition 4 by taking as a base for the topology all intervals of the form
  [a,b){displaystyle [a,b)}
  and for x∈[a,b){displaystyle xin [a,b)}
  by letting μ(x,[a,b))=[x,b){displaystyle mu (x,[a,b))=[x,b)}
  . Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space.
• Any generalised metric is monotonically normal.

Properties[edit]

• Monotone normality is a hereditary property: Every subspace of a monotonically normal space is monotonically normal.
• Every monotonically normal space is completely normal Hausdorff (or T₅).
• Every monotonically normal space is hereditarily collectionwise normal.^[8]
• The image of a monotonically normal space under a continuous closed map is monotonically normal.^[9]
• A compact Hausdorff space
  X{displaystyle X}
  is the continuous image of a compact linearly ordered space if and only if X{displaystyle X}
  is monotonically normal.^[10][3]

References[edit]