Math 2.0

General topology Definition

Continuum

Builds on Metric spaceCompact spaceConnected spaceHausdorff space

Definition

A continuum is a nonempty metric space that is compact and connected. A continuum is nondegenerate if it has more than one point. A subcontinuum of a continuum XX is a subset of XX that is a continuum in the induced metric.

Remarks

The closed interval [0,1][0, 1] is a continuum: it is compact by the Heine–Borel theorem and connected because it is an interval. The set [0,1]∪[2,3][0, 1] \cup [2, 3] is compact but not connected, so it is not a continuum. Some authors define a continuum as a nonempty compact connected Hausdorff space, and call the metric ones metric continua. The intersection of a decreasing sequence of continua is a continuum.

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 spaceconceptConnected space, concept, no evidence levelConnected spaceconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptMetric space, concept, no evidence levelMetric spaceconceptContinuum, definition, no evidence levelContinuumdefinition
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. Connected space concept
  3. Hausdorff space concept
  4. Metric space concept
  5. Continuum definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuum_(topology)
About this page
TagHG4S5HNWXF
URIm2://HG4S5HNWXF
Kinddefinition
WritingW0 stub
MSC54F15
DefinesContinuum
UsesMetric space, Compact space, Connected space, Hausdorff space
Changed2 November 2026

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