Component of a space

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[edit]

[1] |  K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)

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