In mathematics, in the field of

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

X

is said to be collectionwise Hausdorff if given any

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

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.FD Tall
The density topology

Pacific Journal of Mathematics The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisati ...

, 1976 Here a subset

S\subseteq X

being ''discrete'' has the usual meaning of being a discrete space with the

subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...

(i.e., all points of

S

are isolated in

S

).If

X

is T₁ space,

S\subseteq X

being closed and discrete is equivalent to the family of singletons

\

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=\

with the indiscrete topology,

S=\

is discrete but not closed, even though the corresponding family of singletons is a discrete family in

X

Properties

* Every T₁ space that is collectionwise Hausdorff is also Hausdorff. * Every

collectionwise normal In mathematics, a topological space X is called collectionwise normal if for every discrete family ''F'i'' (''i'' ∈ ''I'') of closed subsets of X there exists a pairwise disjoint family of open sets ''U'i'' (''i'' ∈ ''I''), such th ...

space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset

S

X

, every singleton

\

(s\in S)

is closed in

X

and the family of such singletons is a discrete family in

X

.) *

Metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric Metric or metrical may refer t ...

s are collectionwise normal and hence collectionwise Hausdorff.

Remarks

References

{{refbegin Topology Properties of topological spaces