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: ???

gollark: The broader issue is that you have alienated a bunch of the community by imposing the noncompromise earlier.

gollark: As I said, I consider that compromise fine with regards to the specific issue it is actually addressing.

gollark: I don't actually have any authority whatsoever beyond helper powers and control of ██% of the bots here, really.

gollark: Except you just unilaterally came up with that initial statement, so if I wanted to (and I totally do) I could reinterpret that as you demanding some stuff for nothing in return.

References

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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