# Collectionwise normal space

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{{short description|Property of topological spaces stronger than normality}}

In mathematics, a [topological space](/source/topological_space) <math>X</math> is called '''collectionwise normal''' if for every discrete family ''F''<sub>''i''</sub> (''i'' &isin; ''I'') of [closed subset](/source/closed_subset)s of <math>X</math> there exists a [pairwise disjoint](/source/pairwise_disjoint) family of open sets ''U''<sub>''i''</sub> (''i'' &isin; ''I''), such that ''F''<sub>''i''</sub> ⊆ ''U''<sub>''i''</sub>.  Here a family <math>\mathcal{F}</math> of subsets of <math>X</math> is called ''discrete'' when every point of <math>X</math> has a [neighbourhood](/source/Neighbourhood_(mathematics)) that intersects at most one of the sets from  <math>\mathcal{F}</math>.
An equivalent definition<ref>Engelking, Theorem 5.1.17, shows the equivalence between the two definitions (under the assumption of T<sub>1</sub>, but the proof does not use the T<sub>1</sub> property).</ref> of collectionwise normal demands that the above ''U''<sub>''i''</sub> (''i'' &isin; ''I'') themselves form a discrete family, which is ''a priori'' stronger than pairwise disjoint.

Some authors assume that <math>X</math> is also a [T<sub>1</sub> space](/source/T%E2%82%81_space) as part of the definition, but no such assumption is made here.

The property is intermediate in strength between [paracompactness](/source/Paracompact_space) and [normality](/source/Normal_space), and occurs in [metrization theorem](/source/metrization_theorem)s.

==Properties==

*A collectionwise normal space is [collectionwise Hausdorff](/source/collectionwise_Hausdorff).
*A collectionwise normal space is [normal](/source/Normal_space).
*A [Hausdorff](/source/Hausdorff_space) [paracompact space](/source/paracompact_space) is collectionwise normal.{{sfn|Engelking|1989|loc=Theorem 5.1.18}}  In particular, every [metrizable space](/source/metrizable_space) is collectionwise normal.<br>Note: The Hausdorff condition is necessary here, since for example an infinite set with the [cofinite topology](/source/cofinite_topology) is [compact](/source/compact_space), hence paracompact, and T<sub>1</sub>, but is not even normal.
*Every normal [countably compact space](/source/countably_compact_space) (hence every normal compact space) is collectionwise normal.<br>''Proof'': Use the fact that in a countably compact space any discrete family of nonempty subsets is finite.
*An [F<sub>σ</sub>](/source/F-sigma)-set in a collectionwise normal space is also collectionwise normal in the [subspace topology](/source/subspace_topology). In particular, this holds for closed subsets.
*The ''{{visible anchor|Moore metrization theorem}}'' states that a collectionwise normal [Moore space](/source/Moore_space_(topology)) is [metrizable](/source/metrizable).

==Hereditarily collectionwise normal space==

A topological space ''X'' is called '''hereditarily collectionwise normal''' if every subspace of ''X'' with the subspace topology is collectionwise normal.

In the same way that [hereditarily normal space](/source/hereditarily_normal_space)s can be characterized in terms of [separated sets](/source/separated_sets), there is an equivalent characterization for hereditarily collectionwise normal spaces.  A family <math>F_i (i \in I)</math> of subsets of ''X'' is called a '''separated family''' if for every ''i'', we have <math display=inline>F_i \cap \operatorname{cl}(\bigcup_{j \ne i}F_j) = \empty</math>, with cl denoting the [closure](/source/closure_(topology)) operator in ''X'', in other words if the family of <math>F_i</math> is discrete in its union.  The following conditions are equivalent:{{sfn|Engelking|1989|loc=Problem 5.5.1}}
# ''X'' is hereditarily collectionwise normal.
# Every open subspace of ''X'' is collectionwise normal.
# For every separated family <math>F_i</math> of subsets of ''X'', there exists a pairwise disjoint family of open sets <math>U_i (i \in I)</math>, such that <math>F_i \subseteq U_i</math>.

===Examples of hereditarily collectionwise normal spaces===

* Every [linearly ordered topological space](/source/order_topology) (LOTS)<ref>{{cite journal |last1=Steen |first1=Lynn A. |authorlink = Lynn A. Steen|title=A direct proof that a linearly ordered space is hereditarily collectionwise normal |journal=[Proc. Amer. Math. Soc.](/source/Proc._Amer._Math._Soc.) |date=1970 |volume=24 |issue=4 |pages=727–728 |doi=10.1090/S0002-9939-1970-0257985-7|doi-access=free }}</ref><ref>{{Cite journal |last=Cater |first=Frank S. |date=2006 |title=A Simple Proof that a Linearly Ordered Space is Hereditarily and Completely Collectionwise Normal |journal=[Rocky Mountain Journal of Mathematics](/source/Rocky_Mountain_Journal_of_Mathematics) |volume=36 |issue=4 |pages=1149–1151 |doi=10.1216/rmjm/1181069408 |issn=0035-7596 |jstor=44239306 |zbl=1134.54317 |doi-access=free}}</ref>
* Every [generalized ordered space](/source/order_topology) (GO-space)
* Every [metrizable space](/source/metrizable_space).  This follows from the fact that metrizable spaces are collectionwise normal and being metrizable is a hereditary property.
* Every [monotonically normal space](/source/monotonically_normal_space)<ref>{{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 }}</ref>

==Notes==

{{reflist}}

==References==

* {{cite book|last=Engelking|first=Ryszard| author-link=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}

Category:Properties of topological spaces

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Adapted from the Wikipedia article [Collectionwise normal space](https://en.wikipedia.org/wiki/Collectionwise_normal_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Collectionwise_normal_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.