# Continuum (topology)

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**R**^{2}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 <*x*≤ 1 with the segment −1 ≤*y*≤ 1 of the*y*-axis. It is a one-dimensional continuum that is not arcwise connected, and it is locally disconnected at the points along the*y*-axis. - 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.

- An
*n*-cell**R**^{n}. It is contractible and is the simplest example of an*n*-dimensional continuum. - An
*n*-sphere*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 {

- 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.

- If {(

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. ISBN 0-8247-8659-9.

## External links

- Open problems in continuum theory
- Examples in continuum theory
- Continuum Theory and Topological Dynamics, M. Barge and J. Kennedy, in Open Problems in Topology, J. van Mill and G.M. Reed (Editors) Elsevier Science Publishers B.V. (North-Holland), 1990.
- Hyperspacewiki

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