Math 2.0

General topology Definition

Extremally disconnected space

Builds on Topological spaceClosureOpen set

Definition

Let XX be a topological space. The space XX is extremally disconnected if for every open set U⊆XU \subseteq X, the closure U‾\overline{U} is open in XX.

Remarks

Example: a discrete space is extremally disconnected. Every subset is open, so in particular every closure is open.

Non-example: ℝ\R with the usual topology is not extremally disconnected. The set (0,1)(0, 1) is open, and its closure [0,1][0, 1] is not open.

An equivalent condition: for all disjoint open sets U,V⊆XU, V \subseteq X, the closures U‾\overline{U} and V‾\overline{V} are disjoint. Some authors also require XX to be Hausdorff. An extremally disconnected Hausdorff space is totally disconnected.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Closure, concept, no evidence levelClosureconceptOpen set, concept, no evidence levelOpen setconceptTopological space, concept, no evidence levelTopological spaceconceptExtremally disconnected space, definition, no evidence levelExtremally disconnect…definition
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. Closure concept
  2. Open set concept
  3. Topological space concept
  4. Extremally disconnected space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Extremally_disconnected_space
About this page
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URIm2://D14XVCMGX9
Kinddefinition
WritingW0 stub
MSC54G05
DefinesExtremally disconnected space
UsesTopological space, Closure, Open set
Changed2 November 2026

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