Moore space (topology)

In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold: Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (X is a regular Hausdorff space.) There is a countable collection of open covers of X, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C. (X is a developable space.) Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century. Every metrizable space, X, is a Moore space. If {A(n)x} is the open cover of X (indexed by x in X) by all balls of radius 1/n, then the collection of all such open covers as n varies over the positive integers is a development of X. Since all metrizable spaces are normal, all metric spaces are Moore spaces. Moore spaces are a lot like regular spaces and different from normal spaces in the sense that every subspace of a Moore space is also a Moore space. The image of a Moore space under an injective, continuous open map is always a Moore space. (The image of a regular space under an injective, continuous open map is always regular.) Both examples 2 and 3 suggest that Moore spaces are similar to regular spaces. Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable. The Moore plane (also known as the Niemytski space) is an example of a non-metrizable Moore space. Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem. Every locally compact, locally connected normal Moore space is metrizable. This theorem was proved by Reed and Zenor. If , then every separable normal Moore space is metrizable.

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Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. All spaces in this glossary are assumed to be topological spaces unless stated otherwise. Absolutely closed See H-closed Accessible See . Accumulation point See limit point.

Moore space (topology)