Collectionwise normal space

In mathematics, a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called collectionwise normal if for every discrete family Fi (i ∈ I) of <a href="/facts/Closed_subset/upYnRTUj">closed subsets</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 there exists a <a href="/facts/Pairwise_disjoint/nLr8XRVd">pairwise disjoint</a> family of open sets Ui (i ∈ I), such that Fi ⊆ Ui. Here a family 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 of subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called discrete when every point of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 has a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> that intersects at most one of the sets from 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
.
An equivalent definition of collectionwise normal demands that the above Ui (i ∈ I) themselves form a discrete family, which is a priori stronger than pairwise disjoint.
Some authors assume that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is also a <a href="/facts/T%25E2%2582%2581_space/K4exfuf7">T1 space</a> as part of the definition, but no such assumption is made here.
The property is intermediate in strength between <a href="/facts/Paracompact_space/rjTrvF7j">paracompactness</a> and <a href="/facts/Normal_space/t4DLqLCd">normality</a>, and occurs in <a href="/facts/Metrization_theorem/BCLdo5Jf">metrization theorems</a>.

Collectionwise normal space open-in-new

Collectionwise normal space