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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 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.
 
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.
  
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See the article [[Connectivity]] for related concepts and references.
  
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For Vietoris topology, see [[Hyperspace]].
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For Vietoris topology see [[Hyperspace]].
 
  
 
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====References====
 
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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>
 
<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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A ''separation'' in a topological space is a non-trivial open-closed subset.  A space is thus ''disconnected'' if and only if it has a separation, and ''totally separated'' if for any points $x \neq y$ there is a separation containing $x$ and not $y$. 
 
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The equivalence relation on a space $X$ that $x \equiv y$ if and only if for every separation $S$ of $X$ either  $x$ and $y$ are in $S$ or $x$ and $y$ are in $X \setminus S$ defines equivalence classes called ''quasicomponents''.  The quasicomponent of $x$ is the intersection of all open-closed sets of $X$ containing $x$.  If there is only one quasicomponent in $X$ then it is connected.  A space is totally separated if and only if its quasicomponents are all [[singleton]]s.
 
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The equivalence relation on a space $X$ that $x \equiv y$ if and only if there is a connected subspace of $X$ containing $x$ and $y$ defines as classes the ''connected components'': these are the maximal connected subspaces of $X$.  The component of $x$ is the union of all connected subsets of $X$ containing $x$.  A space is ''[[Totally-disconnected space|totally disconnected]]'' if and only if its connected components are all singletons.
 
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See also: [[Extremally-disconnected space]].
 
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====References====
 
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  Steen, Lynn Arthur; Seebach, J.Arthur jun. ''Counterexamples in topology'' (2nd ed.) Springer (1978) ISBN 0-387-90312-7 {{ZBL|0386.54001}}</TD></TR>
 
 
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Latest revision as of 18:42, 12 August 2023

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

See the article Connectivity for related concepts and references.

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=37265
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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