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.

Definition

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

Definition 1

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The space   is T1 and there is a function   that assigns to each ordered pair   of disjoint closed sets in   an open set   such that:

(i)  ;
(ii)   whenever   and  .

Condition (i) says   is a normal space, as witnessed by the function  . Condition (ii) says that   varies in a monotone fashion, hence the terminology monotonically normal. The operator   is called a monotone normality operator.

One can always choose   to satisfy the property

 ,

by replacing each   by  .

Definition 2

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The space   is T1 and there is a function   that assigns to each ordered pair   of separated sets in   (that is, such that  ) an open set   satisfying the same conditions (i) and (ii) of Definition 1.

Definition 3

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The space   is T1 and there is a function   that assigns to each pair   with   open in   and   an open set   such that:

(i)  ;
(ii) if  , then   or  .

Such a function   automatically satisfies

 .

(Reason: Suppose  . Since   is T1, there is an open neighborhood   of   such that  . By condition (ii),  , that is,   is a neighborhood of   disjoint from  . So  .)[5]

Definition 4

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Let   be a base for the topology of  . The space   is T1 and there is a function   that assigns to each pair   with   and   an open set   satisfying the same conditions (i) and (ii) of Definition 3.

Definition 5

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The space   is T1 and there is a function   that assigns to each pair   with   open in   and   an open set   such that:

(i)  ;
(ii) if   and   are open and  , then  ;
(iii) if   and   are distinct points, then  .

Such a function   automatically satisfies all conditions of Definition 3.

Examples

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  • 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   and for   by letting  . 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

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  • 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   is the continuous image of a compact linearly ordered space if and only if   is monotonically normal.[10][3]

References

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  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.