Lemma 5.7.5. Let $f : X \to Y$ be a continuous map of topological spaces. Assume that (a) $f$ is open, (b) all fibres of $f$ are connected. Then $f$ induces a bijection between the sets of connected components of $X$ and $Y$.

Proof. This is a special case of Lemma 5.7.4. $\square$

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