Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space $X$ is said to be collectionwise Hausdorff if given any closed discrete subset of $X$ , 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 $S\subseteq X$ being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of $S$ are isolated in $S$ ). [nb 1]

Properties

Every T₁ space that is collectionwise Hausdorff is also Hausdorff.

Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset $S$ of $X$ , every singleton $\{s\}$ $(s\in S)$ is closed in $X$ and the family of such singletons is a discrete family in $X$ .)

Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.

Remarks

If $X$ is T₁ space, $S\subseteq X$ being closed and discrete is equivalent to the family of singletons $\{\{s\}:s\in S\}$ being a discrete family of subsets of $X$ (in the sense that every point of $X$ has a neighborhood that meets at most one set in the family). If $X$ is not T₁, the family of singletons being a discrete family is a weaker condition. For example, if $X=\{a,b\}$ with the indiscrete topology, $S=\{a\}$ is discrete but not closed, even though the corresponding family of singletons is a discrete family in $X$ .

gollark: A mildly interesting thing they didn't mention in the list (as far as I can see from here) is whether your drive conserves velocity or not. Needing to decelerate a stupid amount if you travel far is relevant to stuff.

gollark: I wonder how long you could safely be in a star's corona, surface or core for...

gollark: Hopefully you won't miss your desired position and fall into the star or something.

gollark: Your stuff is on the scale of *universes*?!

gollark: You would probably want to put most people into constantly moving habitats if there was any likelihood of being attacked.

References

FD Tall, The density topology, Pacific Journal of Mathematics, 1976

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[2] If $X$ is T₁ space, $S\subseteq X$ being closed and discrete is equivalent to the family of singletons $\{\{s\}:s\in S\}$ being a discrete family of subsets of $X$ (in the sense that every point of $X$ has a neighborhood that meets at most one set in the family). If $X$ is not T₁, the family of singletons being a discrete family is a weaker condition. For example, if $X=\{a,b\}$ with the indiscrete topology, $S=\{a\}$ is discrete but not closed, even though the corresponding family of singletons is a discrete family in $X$ .

[1] FD Tall, The density topology, Pacific Journal of Mathematics, 1976