Continuum (topology)

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Nonempty compact connected metric space

title: "Continuum (topology)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["continuum-theory"] description: "Nonempty compact connected metric space" topic_path: "general/continuum-theory" source: "https://en.wikipedia.org/wiki/Continuum_(topology)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Nonempty compact connected metric space ::

In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.

Definitions

  • A continuum that contains more than one point is called nondegenerate.
  • A subset A of a continuum X such that A itself is a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum.
  • A continuum X is homogeneous if for every two points x and y in X, there exists a homeomorphism h: X → X such that h(x) = y.
  • A Peano continuum is a continuum that is locally connected at each point.
  • An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum X is hereditarily indecomposable if every subcontinuum of X is indecomposable.
  • The dimension of a continuum usually means its topological dimension. A one-dimensional continuum is often called a curve.

Examples

  • An arc is a space homeomorphic to the closed interval [0,1]. If h: [0,1] → X is a homeomorphism and h(0) = p and h(1) = q then p and q are called the endpoints of X; one also says that X is an arc from p to q. An arc is the simplest and most familiar type of a continuum. It is one-dimensional, arcwise connected, and locally connected.

  • The topologist's sine curve is a subset of the plane that is the union of the graph of the function f(x) = sin(1/x), 0

  • The Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not a contractible space. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a2/Warsaw_Circle.png" caption="Warsaw circle"] ::

  • An ** n-cell** is a space homeomorphic to the closed ball in the Euclidean space Rn. It is contractible and is the simplest example of an n-dimensional continuum.

  • An ** n-sphere** is a space homeomorphic to the standard n-sphere in the (n + 1)-dimensional Euclidean space. It is an n-dimensional homogeneous continuum that is not contractible, and therefore different from an n-cell.

  • The Hilbert cube is an infinite-dimensional continuum.

  • Solenoids are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.

  • The Sierpinski carpet, also known as the Sierpinski universal curve, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum.

  • The pseudo-arc is a homogeneous hereditarily indecomposable planar continuum.

Properties

There are two fundamental techniques for constructing continua, by means of nested intersections and inverse limits.

:* If {X**n} is a nested family of continua, i.e. X**n ⊇ X**n+1, then their intersection is a continuum.

:* If {(X**n, f**n)} is an inverse sequence of continua X**n, called the coordinate spaces, together with continuous maps f**n: X**n+1 → X**n, called the bonding maps, then its inverse limit is a continuum.

A finite or countable product of continua is a continuum.

References

Sources

  • Sam B. Nadler, Jr, Continuum theory. An introduction. Pure and Applied Mathematics, Marcel Dekker. .

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

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