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 $Y$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: