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Difference between revisions of "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.
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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 union of two non-empty disjoint [[Open-closed set|open-closed subsets]]. A space is connected if and only if every [[Continuous mapping|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 space]]s.
  
  
  
 
====Comments====
 
====Comments====
For Vietoris topology see [[Hyperspace|Hyperspace]].
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For Vietoris topology see [[Hyperspace]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
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====Comments====
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See the article [[Connectivity]] for related concepts and references.

Latest revision as of 14:06, 2 January 2016

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)


Comments

See the article Connectivity for related concepts and references.

How to Cite This Entry:
Connected space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connected_space&oldid=13799
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article