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Monotonically normal space
Normal space in which disjoint neighborhoods of disjoint closed sets can be chosen "monotonically"

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.

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Definition

A topological space X {\displaystyle X} is called monotonically normal if it satisfies any of the following equivalent definitions:1234

Definition 1

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)}}} .

Definition 2

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.

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 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

Definition 4

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.

Definition 5

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.

Examples

  • Every metrizable space is monotonically normal.6
  • Every linearly ordered topological space (LOTS) is monotonically normal.78 This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.9
  • The Sorgenfrey line is monotonically normal.10 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.

Properties

  • 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.11
  • The image of a monotonically normal space under a continuous closed map is monotonically normal.12
  • 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.1314

References

  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. https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf ↩

  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. https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf ↩

  3. 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. https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf ↩

  4. Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm ↩

  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. https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf ↩

  6. Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm ↩

  7. Heath, Lutzer, Zenor, Theorem 5.3 ↩

  8. Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm ↩

  9. 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. /wiki/Eric_van_Douwen ↩

  10. Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm ↩

  11. Heath, Lutzer, Zenor, Theorem 3.1 ↩

  12. Heath, Lutzer, Zenor, Theorem 2.6 ↩

  13. Rudin, Mary Ellen (2001). "Nikiel's conjecture" (PDF). Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8. https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf ↩

  14. 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. https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf ↩