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)