Continuous maps between compact Hausdorff spaces and their induced maps on their space of continuous real-valued functions.

Question

Let $X,Y$ be two compact Hausdorff spaces and let $ \alpha: X \to Y$ be continuous onto map. Let $\alpha^{*}$ be the induced map from $C(Y)$ to $C(X)$ by mapping any $f \in C(Y)$ to $f \circ \alpha \in C(X)$.

---

First, is it true that the map $\alpha^{*}$ is surjective?

Second, if my first proposition is not always true, then if we substitute "Compact Hausdorff space" with "extremally disconnected compact Hausdorff space", is my first proposition always true? Why?

Thanks.

Answer 1

Suppose that $Y$ is a point and $X$ is $S^n$ for example, $f\circ \alpha$ is always constant, there are non constant functions on $S^n$.