Math 2.0

General topology Definition

Moore space

Builds on Topological spaceOpen coverRegular spaceHausdorff space

Definition

Let XX be a topological space. For a family ๐’ข\mathcal{G} of subsets of XX and a point xโˆˆXx \in X, the star of xx in ๐’ข\mathcal{G} is stโก(x,๐’ข)=โ‹ƒ{Gโˆˆ๐’ข:xโˆˆG}\operatorname{st}(x, \mathcal{G}) = \bigcup \set{G \in \mathcal{G} : x \in G}. A development of XX is a sequence ๐’ข1,๐’ข2,โ€ฆ\mathcal{G}_1, \mathcal{G}_2, \dots of open covers of XX with the following property. For every xโˆˆXx \in X and every open set UU with xโˆˆUx \in U, there is nn with stโก(x,๐’ขn)โŠ†U\operatorname{st}(x, \mathcal{G}_n) \subseteq U. The space XX is developable if it has a development. A Moore space is a developable space that is regular and Hausdorff.

Remarks

Every metric space (X,d)(X, d) is a Moore space. Let ๐’ขn\mathcal{G}_n be the cover by all open balls of radius 1/n1/n. Every ball of ๐’ขn\mathcal{G}_n that contains xx lies in the ball B(x,2/n)B(x, 2/n), so stโก(x,๐’ขn)โŠ†B(x,2/n)\operatorname{st}(x, \mathcal{G}_n) \subseteq B(x, 2/n), and this lies in UU for nn large. A set XX with two points and the indiscrete topology is developable, with ๐’ขn={X}\mathcal{G}_n = \set{X} for all nn, because XX is the only open set containing a point. It is not a Moore space, because it is not Hausdorff. Some authors do not include regularity or the Hausdorff condition in the definition, so the word should be checked in each text.

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Hausdorff space, concept, no evidence levelHausdorff spaceconceptOpen cover, concept, no evidence levelOpen coverconceptRegular space, concept, no evidence levelRegular spaceconceptTopological space, concept, no evidence levelTopological spaceconceptMoore space, definition, no evidence levelMoore spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Hausdorff space concept
  2. Open cover concept
  3. Regular space concept
  4. Topological space concept
  5. Moore space definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Moore_space_(topology)
About this page
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Kinddefinition
WritingW0 stub
MSC54E30
DefinesMoore space
UsesTopological space, Open cover, Regular space, Hausdorff space
Changed2 November 2026

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