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Monotonically normal space - Wikipedia

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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.

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.

1. ^ Heath, R. W.; Lutzer, D. J.; Zenor, P. L. (April 1973). "Monotonically Normal Spaces" (PDF). Transactions of the American Mathematical Society. 178: 481–493. doi:10.2307/1996713. JSTOR 1996713.
2. ^ Borges, Carlos R. (March 1973). "A Study of Monotonically Normal Spaces" (PDF). Proceedings of the American Mathematical Society. 38 (1): 211–214. doi:10.2307/2038799. JSTOR 2038799.
3. ^ ^{a b} Bennett, Harold; Lutzer, David (2015). "Mary Ellen Rudin and monotone normality" (PDF). Topology and Its Applications. 195: 50–62. doi:10.1016/j.topol.2015.09.021.
4. ^ ^{a b c d} Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist.
5. ^ Zhang, Hang; Shi, Wei-Xue (2012). "Monotone normality and neighborhood assignments" (PDF). Topology and Its Applications. 159 (3): 603–607. doi:10.1016/j.topol.2011.10.007.
6. ^ Heath, Lutzer, Zenor, Theorem 5.3
7. ^ van Douwen, Eric K. (September 1985). "Horrors of Topology Without AC: A Nonnormal Orderable Space" (PDF). Proceedings of the American Mathematical Society. 95 (1): 101–105. doi:10.2307/2045582. JSTOR 2045582.
8. ^ Heath, Lutzer, Zenor, Theorem 3.1
9. ^ Heath, Lutzer, Zenor, Theorem 2.6
10. ^ Rudin, Mary Ellen (2001). "Nikiel's conjecture" (PDF). Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8.

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