Difference between revisions of "Connected space"
From Encyclopedia of Mathematics
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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. | + | 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 (if contains more than one point), although there also exist countable connected Hausdorff spaces. |
Revision as of 17:57, 30 September 2013
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 (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://encyclopediaofmath.org/index.php?title=Connected_space&oldid=30582
Connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connected_space&oldid=30582
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article