Collectionwise Hausdorff space
Encyclopedia
In mathematics
, in the field of topology
, a topological space
is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points. Every T1 space
which is collectionwise Hausdorff is also Hausdorff
.
Metrizable spaces are collectionwise normal spaces
and are hence, in particular, collectionwise Hausdorff.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be collectionwise Hausdorff if given any closed discrete collection of points in the topological space, there are pairwise disjoint open sets containing the points. Every T1 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
which is collectionwise Hausdorff is also Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
Metrizable spaces are collectionwise normal spaces
Collectionwise normal
In mathematics, a topological space X is called collectionwise normal if for every discrete family Fi of closed subsets of X there exists a pairwise disjoint family of open sets Ui , such that Fi ⊂ Ui...
and are hence, in particular, collectionwise Hausdorff.