The property is therefore intermediate in strength between paracompactness and normality.Every metrizable space (i. e., every topological space that is homeomorphic to a metric space) is collectionwise normal.The Moore metrisation theorem states that every collectionwise normal Moore space is metrizable.…

Every metrizable space (i. e., every topological space that is homeomorphic to a metric space) is collectionwise normal.The Moore metrisation theorem states that every collectionwise normal Moore space is metrizable.…

Moore himself proved the theorem that a collectionwise normal Moore space is metrizable, so strengthening normality is another way to settle the matter.

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.…

Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem.It is a common tool to prove other metrization theorems, e.g. the Moore metrization theorem: a collectionwise normal, Moore space is metrizable, is a direct consequence.Unlike the Urysohn's metrization theorem which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable.…

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.…

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.The property is therefore intermediate in strength between paracompactness and normality.…