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.…

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.…

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.…

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.…