Collectionwise Hausdorff space
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In mathematics, in the field of topology, a topological space [math]\displaystyle{ X }[/math] is said to be collectionwise Hausdorff if given any closed discrete subset of [math]\displaystyle{ X }[/math], there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.[1] Here a subset [math]\displaystyle{ S\subseteq X }[/math] being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of [math]\displaystyle{ S }[/math] are isolated in [math]\displaystyle{ S }[/math]).[nb 1]
Properties
- Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X }[/math], every singleton [math]\displaystyle{ \{s\} }[/math] [math]\displaystyle{ (s\in S) }[/math] is closed in [math]\displaystyle{ X }[/math] and the family of such singletons is a discrete family in [math]\displaystyle{ X }[/math].)
- Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.
Remarks
- ↑ If [math]\displaystyle{ X }[/math] is T1 space, [math]\displaystyle{ S\subseteq X }[/math] being closed and discrete is equivalent to the family of singletons [math]\displaystyle{ \{\{s\}:s\in S\} }[/math] being a discrete family of subsets of [math]\displaystyle{ X }[/math] (in the sense that every point of [math]\displaystyle{ X }[/math] has a neighborhood that meets at most one set in the family). If [math]\displaystyle{ X }[/math] is not T1, the family of singletons being a discrete family is a weaker condition. For example, if [math]\displaystyle{ X=\{a,b\} }[/math] with the indiscrete topology, [math]\displaystyle{ S=\{a\} }[/math] is discrete but not closed, even though the corresponding family of singletons is a discrete family in [math]\displaystyle{ X }[/math].
References
- ↑ FD Tall, The density topology, Pacific Journal of Mathematics, 1976
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