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Component of a space

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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
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article