View source for Component of a space

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>.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR></table>