Connected fibres and quotient implies connected space

Let $𝑋$ be a topological space, $𝑓 : 𝑋 ↠ 𝑆$ a surjective function, where $𝑆$ is given the Quotient topology. If $𝑆$ is a connected space and each of the fibres of $𝑓$ (i.e. equivalence classes) are connected subspaces of $𝑋$ , then $𝑋$ is connected.

Proof

Let $𝑔 : 𝑋 \to {0, 1}$ . Since the fibres of $𝑓$ are connected, $𝑔$ must be constant on each fibre. Thus there exists $―― 𝑔 : 𝑌 \to {0, 1}$ so that $𝑔 = ―― 𝑔 𝑓$ . But $𝑌$ is connected so $―― 𝑔$ must be constant, so $𝑔$ is also constant. Therefore $𝑋$ is connected.

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