Lemma 5.8.5. Let $f : X \to Y$ be a surjective, continuous map of topological spaces. If $X$ has a finite number, say $n$, of irreducible components, then $Y$ has $\leq n$ irreducible components.
Proof. Say $X_1, \ldots , X_ n$ are the irreducible components of $X$. By Lemmas 5.8.2 and 5.8.3 the closure $Y_ i \subset Y$ of $f(X_ i)$ is irreducible. Since $f$ is surjective, we see that $Y$ is the union of the $Y_ i$. We may choose a minimal subset $I \subset \{ 1, \ldots , n\} $ such that $Y = \bigcup _{i \in I} Y_ i$. Then we may apply Lemma 5.8.4 to see that the $Y_ i$ for $i \in I$ are the irreducible components of $Y$. $\square$
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