Connected space

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A topological space that cannot be represented as the sum of two parts separated from one another, or, more precisely, as the sum 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 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, although there also exist countable connected Hausdorff spaces.


For Vietoris topology see Hyperspace.


[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
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Connected space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article