# Connected space

A topological space that cannot be represented as the sum of two parts separated from one another, or, more precisely, as the union of two non-empty disjoint open-closed subsets. A space is connected if and only if every continuous real-valued function on it takes all intermediate values. The continuous image of a connected space, the topological product of connected spaces, and the space of closed subsets of a connected space in the Vietoris topology are connected spaces. Every connected completely-regular space has cardinality not less than the cardinality of the continuum (if contains more than one point), although there also exist countable connected Hausdorff spaces.

#### Comments

For Vietoris topology see Hyperspace.

#### References

[a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

**How to Cite This Entry:**

Connected space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Connected_space&oldid=31024