Monotonically normal space

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

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 A\subseteq G(A,B)\subseteq {\overline {G(A,B)}}\subseteq X\setminus B} ;

(ii) G ( A , B ) ⊆ G ( A ′ , B ′ ) {\displaystyle G(A,B)\subseteq G(A',B')} whenever A ⊆ A ′ {\displaystyle A\subseteq 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)}}} .

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 A\cap {\overline {B}}=B\cap {\overline {A}}=\emptyset } ) an open set G ( A , B ) {\displaystyle G(A,B)} satisfying the same conditions (i) and (ii) of Definition 1.

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 x\in U} an open set μ ( x , U ) {\displaystyle \mu (x,U)} such that:

(i) x ∈ μ ( x , U ) {\displaystyle x\in \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 x\in V} or y ∈ U {\displaystyle y\in U} .

Such a function μ {\displaystyle \mu } automatically satisfies

x ∈ μ ( x , U ) ⊆ μ ( x , U ) ¯ ⊆ U {\displaystyle x\in \mu (x,U)\subseteq {\overline {\mu (x,U)}}\subseteq U} .

(Reason: Suppose y ∈ X ∖ U {\displaystyle y\in X\setminus U} . Since X {\displaystyle X} is T₁, there is an open neighborhood V {\displaystyle V} of y {\displaystyle y} such that x ∉ V {\displaystyle x\notin 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 y\notin {\overline {\mu (x,U)}}} .)^[5]

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 U\in {\mathcal {B}}} and x ∈ U {\displaystyle x\in U} an open set μ ( x , U ) {\displaystyle \mu (x,U)} satisfying the same conditions (i) and (ii) of Definition 3.

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 x\in U} an open set μ ( x , U ) {\displaystyle \mu (x,U)} such that:

(i) x ∈ μ ( x , U ) {\displaystyle x\in \mu (x,U)} ;

(ii) if U {\displaystyle U} and V {\displaystyle V} are open and x ∈ U ⊆ V {\displaystyle x\in U\subseteq 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,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset } .

Such a function μ {\displaystyle \mu } automatically satisfies all conditions of Definition 3.

• 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 x\in [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.

• 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]