Math 2.0

General topology Definition

Countably compact space

Builds on Topological spaceOpen coverCountable setCompact spaceSequential compactnessLimit point compact spaceT1 space

Definition

A topological space XX is countably compact when every countable open cover of XX has a finite subcover.

Remarks

The following are equivalent for any topological space XX:

  1. XX is countably compact;
  2. every sequence in XX has a cluster point, that is, a point xx such that each neighbourhood of xx contains xnx_n for infinitely many nn;
  3. every countable family of closed sets with the finite intersection property has nonempty intersection.

Every compact space and every sequentially compact space is countably compact. Every countably compact space is limit point compact, and the converse holds for T1 spaces. A countably compact space that is also Lindelöf is compact.

Example: [0,1][0, 1] is countably compact, since it is compact by the Heine–Borel theorem. Example: the space [0,ω1)[0, \omega_1) of countable ordinals with the order topology is countably compact but not compact.

Non-example: ℝ\R is not countably compact. The countable open cover by the intervals (−n,n)(-n, n) for n≥1n \ge 1 has no finite subcover.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Compact space, concept, no evidence levelCompact spaceconceptCountable set, concept, no evidence levelCountable setconceptLimit point compact space, concept, no evidence levelLimit point compact s…conceptOpen cover, concept, no evidence levelOpen coverconceptSequential compactness, concept, no evidence levelSequential compactnessconceptT1 space, concept, no evidence levelT1 spaceconceptTopological space, concept, no evidence levelTopological spaceconceptCountably compact space, definition, no evidence levelCountably compact spa…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.
  • no evidence level
  1. Compact space concept
  2. Countable set concept
  3. Limit point compact space concept
  4. Open cover concept
  5. Sequential compactness concept
  6. T1 space concept
  7. Topological space concept
  8. Countably compact space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Countably_compact_space
About this page
TagDV8GH6HJJ8
URIm2://DV8GH6HJJ8
Kinddefinition
WritingW0 stub
MSC54D30
DefinesCountably compact space
UsesTopological space, Open cover, Countable set, Compact space, Sequential compactness, Limit point compact space, T1 space
Changed2 November 2026

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