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

A topological space $X$ is called monotonically normal if it satisfies any of the following equivalent definitions:

Definition 1

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

(i)

A \subseteq G (A, B) \subseteq \overline{G (A, B)} \subseteq X ∖ B

;

(ii)

G (A, B) \subseteq G (A^{'}, B^{'})

whenever

A \subseteq A^{'}

and

B^{'} \subseteq B

.

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

One can always choose $G$ to satisfy the property

G (A, B) \cap G (B, A) = ∅︀

,

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

Definition 2

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

Definition 3

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

(i)

x \in μ (x, U)

;

(ii) if

μ (x, U) \cap μ (y, V) \neq ∅︀

, then

x \in V

or

y \in U

.

Such a function $μ$ automatically satisfies

x \in μ (x, U) \subseteq \overline{μ (x, U)} \subseteq U

.

(Reason: Suppose $y \in X ∖ U$ . Since $X$ is T₁, there is an open neighborhood $V$ of $y$ such that $x \notin V$ . By condition (ii), $μ (x, U) \cap μ (y, V) = ∅︀$ , that is, $μ (y, V)$ is a neighborhood of $y$ disjoint from $μ (x, U)$ . So $y \notin \overline{μ (x, U)}$ .)

Definition 4

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

Definition 5

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

(i)

x \in μ (x, U)

;

(ii) if

U

and

V

are open and

x \in U \subseteq V

, then

μ (x, U) \subseteq μ (x, V)

;

(iii) if

x

and

y

are distinct points, then

μ (x, X ∖ {y}) \cap μ (y, X ∖ {x}) = ∅︀

.

Such a function $μ$ automatically satisfies all conditions of Definition 3.

Examples

Every metrizable space is monotonically normal.
Every linearly ordered topological space (LOTS) is monotonically normal. This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.
The Sorgenfrey line is monotonically normal. This follows from Definition 4 by taking as a base for the topology all intervals of the form $[a, b)$ and for $x \in [a, b)$ by letting $μ (x, [a, b)) = [x, b)$ . 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.
The image of a monotonically normal space under a continuous closed map is monotonically normal.
A compact Hausdorff space $X$ is the continuous image of a compact linearly ordered space if and only if $X$ is monotonically normal.

References

↑ 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.
1 2 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.
1 2 3 4 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.

[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.

[Rudin-3] 1 2 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.

[Brandsma-4] 1 2 3 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 (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.