Monotonically normal space

Separation axioms in topological spaces
Kolmogorov classification
T0	(Kolmogorov)
T1	(Fréchet)
T2	(Hausdorff)
T2½	(Urysohn)
completely T2	(completely Hausdorff)
T3	(regular Hausdorff)
T3½	(Tychonoff)
T4	(normal Hausdorff)
T5	(completely normal  Hausdorff)
T6	(perfectly normal  Hausdorff)
History

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

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

Definition 1

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

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

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

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

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

• 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

• 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]}

See also

• Collectionwise normal space  – Property of topological spaces stronger than normality
• Locally normal space
• Paranormal space
• Separation axiom  – Axioms in topology defining notions of "separation"

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.
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. 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. 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.