Math 2.0

Order, lattices, ordered algebraic structures Definition

Stone space

Builds on Topological spaceCompact spaceHausdorff spaceTotally disconnected space

Definition

A Stone space is a topological space XX such that:

  1. XX is compact.
  2. XX is Hausdorff.
  3. XX is totally disconnected: its only connected subsets with at least one point are the singletons.

Remarks

For a compact Hausdorff space, clause 3 is equivalent to having a basis of clopen sets, that is, to being zero-dimensional.

Example: the Cantor space {0,1}ℕ\set{0, 1}^{\N} with the product topology. It is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces. Two different sequences differ at some index nn, and the set of sequences with a fixed nn-th term is clopen, so no connected set contains both.

Non-example: the interval [0,1][0, 1] is compact and Hausdorff but connected, so clause 3 fails. The space ℚ\Q is Hausdorff and totally disconnected but not compact, so clause 1 fails.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptTopological space, concept, no evidence levelTopological spaceconceptTotally disconnected space, concept, no evidence levelTotally disconnected …conceptStone space, definition, no evidence levelStone 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. Compact space concept
  2. Hausdorff space concept
  3. Topological space concept
  4. Totally disconnected space concept
  5. Stone space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stone_space
About this page
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Kinddefinition
WritingW0 stub
MSC06E15, 54D30
DefinesStone space
UsesTopological space, Compact space, Hausdorff space, Totally disconnected space
Changed2 November 2026

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