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 \( X \).

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