Continuous Image of Normal Space is Normal

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.

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]

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 ${\displaystyle U}$ 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]

References [edit]

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: 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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Source: https://en.wikipedia.org/wiki/Monotonically_normal_space