Difference between revisions of "Component of a space"
From Encyclopedia of Mathematics
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| − | A connected subset <math> C </math> of a topological space <math> X </math> with the following property: If <math>C_1 \subset X</math> is a connected subset such that <math> C \subset C_1 </math>, then <math> C = C_1 </math>. The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If <math> C </math> is a component of a space <math> X </math> and <math> C \subset Y \subset X </math>, then <math> C </math> is a component of <math> Y </math>. If <math> \mathit{f}:X \to Y </math> is a monotone continuous mapping onto, then <math> C </math> is a component of <math> Y </math> if and only if <math> \mathit{f}^{-1}(C) </math> is a component of <math> X </math>. | + | A component of a space is a [[Connected set|connected]] subset <math> C </math> of a [[topological space]] <math> X </math> with the following property: If <math>C_1 \subset X</math> is a connected subset such that <math> C \subset C_1 </math>, then <math> C = C_1 </math>. The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If <math> C </math> is a component of a space <math> X </math> and <math> C \subset Y \subset X </math>, then <math> C </math> is a component of <math> Y </math>. If <math> \mathit{f}:X \to Y </math> is a [[Monotone mapping|monotone]] [[Continuous mapping|continuous]] mapping onto, then <math> C </math> is a component of <math> Y </math> if and only if <math> \mathit{f}^{-1}(C) </math> is a component of <math> X </math>. |
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | ||
Latest revision as of 22:41, 25 March 2011
A component of a space is a connected subset \( C \) of a topological space \( X \) with the following property: If \(C_1 \subset X\) is a connected subset such that \( C \subset C_1 \), then \( C = C_1 \). The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If \( C \) is a component of a space \( X \) and \( C \subset Y \subset X \), then \( C \) is a component of \( Y \). If \( \mathit{f}:X \to Y \) is a monotone continuous mapping onto, then \( C \) is a component of \( Y \) if and only if \( \mathit{f}^{-1}(C) \) is a component of \( X \).
References
| [1] | K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French) |
How to Cite This Entry:
Component of a space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Component_of_a_space&oldid=19333
Component of a space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Component_of_a_space&oldid=19333
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article