Many authors assume that X is also a T 1 space as part of the definition, i. e., for every pair of distinct points, each has an open neighborhood not containing the other.A collectionwise normal T 1 space is a collectionwise Hausdorff space.Every collectionwise normal space is normal (i. e., any two disjoint closed sets can be separated by neighbourhoods), and every paracompact space (i. e., every topological space in which every open cover admits a locally finite open refinement) is collectionwise normal.…

A closed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most one point from S. Every T1 space which is collectionwise Hausdorff is also Hausdorff.Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.…

In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras.An extremally disconnected first countable collectionwise Hausdorff space must be discrete.In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).…

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.A closed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most one point from S. Every T1 space which is collectionwise Hausdorff is also Hausdorff.…

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.A closed discrete set S of a topology X is one where every point of X has a neighborhood that intersects at most one point from S. Every T1 space which is collectionwise Hausdorff is also Hausdorff.Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.…