Collectionwise normal

In mathematics, a topological space ${\displaystyle X}$ is called collectionwise normal if for every discrete family F_i (i ∈ I) of closed subsets of ${\displaystyle X}$ there exists a pairwise disjoint family of open sets U_i (i ∈ I), such that F_i ⊂ U_i. A family ${\displaystyle {\mathcal {F}}}$ of subsets of ${\displaystyle X}$ is called discrete when every point of ${\displaystyle X}$ has a neighbourhood that intersects at most one of the sets from ${\displaystyle {\mathcal {F}}}$. An equivalent definition demands that the above U_i (i ∈ I) are themselves a discrete family, which is stronger than pairwise disjoint.

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