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:^[1]^[2]^[3]^[4]

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\setminus 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)=\emptyset

,

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

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 $A\cap {\overline {B}}=B\cap {\overline {A}}=\emptyset$ ) 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 $\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\in U$ an open set $\mu (x,U)$ such that:

(i)

x\in \mu (x,U)

;

(ii) if

\mu (x,U)\cap \mu (y,V)\neq \emptyset

, then

x\in V

or

y\in U

.

Such a function $\mu$ automatically satisfies

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

.

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

Definition 4

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

Definition 5

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

(i)

x\in \mu (x,U)

;

(ii) if

U

and

V

are open and

x\in U\subseteq V

, then

\mu (x,U)\subseteq \mu (x,V)

;

(iii) if

x

and

y

are distinct points, then

\mu (x,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset

.

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

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.
^ ^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.

[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] 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] 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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Fahrer / Team für die Suche eingeben: