Monotonically normal space

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

Definition 1#edit

The space ${\displaystyle X}$ is T₁ and there is a function ${\displaystyle G}$ that assigns to each ordered pair ${\displaystyle (A,B)}$ of disjoint closed sets in ${\displaystyle X}$ an open set ${\displaystyle G(A,B)}$ such that:

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

(ii) ${\displaystyle G(A,B)\subseteq G(A',B')}$ whenever ${\displaystyle A\subseteq A'}$ and ${\displaystyle B'\subseteq B}$ .

Condition (i) says ${\displaystyle X}$ is a normal space, as witnessed by the function ${\displaystyle G}$ . Condition (ii) says that ${\displaystyle G(A,B)}$ varies in a monotone fashion, hence the terminology monotonically normal. The operator ${\displaystyle G}$ is called a monotone normality operator.

One can always choose ${\displaystyle G}$ to satisfy the property

${\displaystyle G(A,B)\cap G(B,A)=\emptyset }$ ,

by replacing each ${\displaystyle G(A,B)}$ by ${\displaystyle G(A,B)\setminus {\overline {G(B,A)$ .

Definition 2#edit

The space ${\displaystyle X}$ is T₁ and there is a function ${\displaystyle G}$ that assigns to each ordered pair ${\displaystyle (A,B)}$ of separated sets in ${\displaystyle X}$ (that is, such that ${\displaystyle A\cap {\overline {B=B\cap {\overline {A=\emptyset }$ ) an open set ${\displaystyle G(A,B)}$ satisfying the same conditions (i) and (ii) of Definition 1.

Definition 3#edit

The space ${\displaystyle X}$ is T₁ and there is a function ${\displaystyle \mu }$ that assigns to each pair ${\displaystyle (x,U)}$ with ${\displaystyle U}$ open in ${\displaystyle X}$ and ${\displaystyle x\in U}$ an open set ${\displaystyle \mu (x,U)}$ such that:

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

(ii) if ${\displaystyle \mu (x,U)\cap \mu (y,V)\neq \emptyset }$ , then ${\displaystyle x\in V}$ or ${\displaystyle y\in U}$ .

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

${\displaystyle x\in \mu (x,U)\subseteq {\overline {\mu (x,U)\subseteq U}$ .

(Reason: Suppose ${\displaystyle y\in X\setminus U}$ . Since ${\displaystyle X}$ is T₁, there is an open neighborhood ${\displaystyle V}$ of ${\displaystyle y}$ such that ${\displaystyle x\notin V}$ . By condition (ii), ${\displaystyle \mu (x,U)\cap \mu (y,V)=\emptyset }$ , that is, ${\displaystyle \mu (y,V)}$ is a neighborhood of ${\displaystyle y}$ disjoint from ${\displaystyle \mu (x,U)}$ . So ${\displaystyle y\notin {\overline {\mu (x,U)$ .)^[5]

Definition 4#edit

Let ${\displaystyle {\mathcal {B$ be a base for the topology of ${\displaystyle X}$ . The space ${\displaystyle X}$ is T₁ and there is a function ${\displaystyle \mu }$ that assigns to each pair ${\displaystyle (x,U)}$ with ${\displaystyle U\in {\mathcal {B$ and ${\displaystyle x\in U}$ an open set ${\displaystyle \mu (x,U)}$ satisfying the same conditions (i) and (ii) of Definition 3.

Definition 5#edit

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

(ii) if ${\displaystyle U}$ and ${\displaystyle V}$ are open and ${\displaystyle x\in U\subseteq V}$ , then ${\displaystyle \mu (x,U)\subseteq \mu (x,V)}$ ;

(iii) if ${\displaystyle x}$ and ${\displaystyle y}$ are distinct points, then ${\displaystyle \mu (x,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset }$ .

Such a function ${\displaystyle \mu }$ 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 ${\displaystyle [a,b)}$ and for ${\displaystyle x\in [a,b)}$ by letting ${\displaystyle \mu (x,[a,b))=[x,b)}$ . 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 ${\displaystyle X}$ is the continuous image of a compact linearly ordered space if and only if ${\displaystyle X}$ is monotonically normal.^[10]^[3]

References#edit

^ 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.
^ 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.
^ 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.
^ a b c d Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist.
^ 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.
^ Heath, Lutzer, Zenor, Theorem 5.3
^ 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.
^ Heath, Lutzer, Zenor, Theorem 3.1
^ Heath, Lutzer, Zenor, Theorem 2.6
^ Rudin, Mary Ellen (2001). "Nikiel's conjecture" (PDF). Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8.

Categories Properties of topological spaces

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