Monotonically normal space

A topological space $X$ is called monotonically normal if it satisfies any of the following equivalent definitions:{{cite journal |last1=Heath |first1=R. W. |last2=Lutzer |first2=D. J. |last3=Zenor |first3=P. L. |date=April 1973 |title=Monotonically Normal Spaces |journal=Transactions of the American Mathematical Society |volume=178 |pages=481–493 |url=https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf |doi=10.2307/1996713|jstor=1996713 |doi-access=free }}{{cite journal |last=Borges |first=Carlos R. |date=March 1973 |title=A Study of Monotonically Normal Spaces |journal=Proceedings of the American Mathematical Society |volume=38 |number=1 |pages=211–214 |url=https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf |doi=10.2307/2038799|jstor=2038799 |doi-access=free }}{{cite journal |last1=Bennett |first1=Harold |last2=Lutzer |first2=David |title=Mary Ellen Rudin and monotone normality |journal=Topology and Its Applications |date=2015 |volume=195 |pages=50–62 |doi=10.1016/j.topol.2015.09.021 |url=https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf}}{{cite web |last1=Brandsma |first1=Henno |title=monotone normality, linear orders and the Sorgenfrey line |url=http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm |website=Ask a Topologist}}

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

:(ii) $G(A,B)\backslash subseteq\; G(A\text{'},B\text{'})$ whenever $A\backslash subseteq\; A\text{'}$ and $B\text{'}\backslash subseteq\; B$.

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

:$G(A,B)\backslash cap\; G(B,A)=\backslash emptyset$,

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

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 $A\backslash cap\backslash overline\{B\}=B\backslash cap\backslash overline\{A\}=\backslash emptyset$) an open set $G(A,B)$ satisfying the same conditions (i) and (ii) of Definition 1.

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

:(i) $x\backslash in\backslash mu(x,U)$;

:(ii) if $\backslash mu(x,U)\backslash cap\backslash mu(y,V)\backslash ne\backslash emptyset$, then $x\backslash in\; V$ or $y\backslash in\; U$.

Such a function $\backslash mu$ automatically satisfies

:$x\backslash in\backslash mu(x,U)\backslash subseteq\backslash overline\{\backslash mu(x,U)\}\backslash subseteq\; U$.

(Reason: Suppose $y\backslash in\; X\backslash setminus\; U$. Since $X$ is T₁, there is an open neighborhood $V$ of $y$ such that $x\backslash notin\; V$. By condition (ii), $\backslash mu(x,U)\backslash cap\backslash mu(y,V)=\backslash emptyset$, that is, $\backslash mu(y,V)$ is a neighborhood of $y$ disjoint from $\backslash mu(x,U)$. So $y\backslash notin\backslash overline\{\backslash mu(x,U)\}$.){{cite journal |last1=Zhang |first1=Hang |last2=Shi |first2=Wei-Xue |title=Monotone normality and neighborhood assignments |journal=Topology and Its Applications |date=2012 |volume=159 |issue=3 |pages=603–607 |doi=10.1016/j.topol.2011.10.007 |url=https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf}}

Let $\backslash mathcal\{B\}$ be a base for the topology of $X$.

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

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

:(i) $x\backslash in\backslash mu(x,U)$;

:(ii) if $U$ and $V$ are open and $x\backslash in\; U\backslash subseteq\; V$, then $\backslash mu(x,U)\backslash subseteq\backslash mu(x,V)$;

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

Such a function $\backslash mu$ automatically satisfies all conditions of Definition 3.

• Every metrizable space is monotonically normal.
• Every linearly ordered topological space (LOTS) is monotonically normal.Heath, Lutzer, Zenor, Theorem 5.3 This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.{{cite journal |last=van Douwen |first=Eric K. |authorlink=Eric van Douwen |date=September 1985 |title=Horrors of Topology Without AC: A Nonnormal Orderable Space |journal=Proceedings of the American Mathematical Society |volume=95 |number=1 |pages=101–105 |url=https://www.ams.org/proc/1985-095-01/S0002-9939-1985-0796455-5/S0002-9939-1985-0796455-5.pdf |doi=10.2307/2045582|jstor=2045582 }}
• 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 $x\backslash in[a,b)$ by letting $\backslash mu(x,[a,b))=[x,b)$. 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.

• 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.Heath, Lutzer, Zenor, Theorem 3.1
• The image of a monotonically normal space under a continuous closed map is monotonically normal.Heath, Lutzer, Zenor, Theorem 2.6
• A compact Hausdorff space $X$ is the continuous image of a compact linearly ordered space if and only if $X$ is monotonically normal.{{cite journal |last1=Rudin |first1=Mary Ellen |title=Nikiel's conjecture |journal=Topology and Its Applications |date=2001 |volume=116 |issue=3 |pages=305–331 |doi=10.1016/S0166-8641(01)00218-8 |url=https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf}}

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Category:Properties of topological spaces