Blowup of $\mathbb{P}^n$ at a point is irreducible

Question

The blowup of $\mathbb{P}^n$ at a point is irreducible.

This seems clear intuitively, but I'm not sure how to prove it. Thoughts?

Answer 1

The isomorphism between $\mathbb{P}^n - {0}$ and the blowup without the exceptional divisor (call that set $U$) gives a continuous map from $\mathbb{P}^n - {0}$ to $U$. $\mathbb{P}^n - {0}$ is irreducible, so its image under a continuous map, $U$, is also irreducible. $U$ is dense in the blowup, the closure of a irreducible set is irreducible and thus the blowup is irreducible.