## collectionwise

10 examples (0.01 sec)
• 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.
• A collectionwise normal T 1 space is a collectionwise Hausdorff space.
• Metrizable spaces are collectionwise normal spaces and are hence, in particular, collectionwise Hausdorff.
• An extremally disconnected first countable collectionwise Hausdorff space must be discrete.
• 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.
• 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.
• 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.
• 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.