Why is the zero locus of an homogeneous polynomial a closed set in the projective space?

Question

I am trying to prove that if I consider the zero locus of an homogeneous polynomial in the real or complex projective space (though I think this applies to all projective spaces over a field), then it is a closed set. Now, I get that the zero locus is well-defined because I have an homogeneous polynomial, however I can't prove it is a closed set. Let's say I have $P^{n}(\mathbb{R})$. If I consider the counterimage of 0 in the projective space, let's say $A$, proving that it is closed would mean to prove that $\pi ^{-1} (A)$ is closed in $\mathbb{R}^{n+1}/R~$, where $R$ is the equivalence relation. I can't prove this, though.

Answer 1

The counterimage of $A$ in $\mathbb{R}^{n+1} - \{ 0 \}$ is the intersection of $\mathbb{R}^{n+1}-\{0\}$ with the zero locus $f^{-1}(0)$ of the homogeneous polynomial $f$, seen as a function $$f \colon \mathbb{R}^{n+1} \to \mathbb{R}$$ on $\mathbb{R}^{n+1}$. Such a zero locus is obviously closed in $\mathbb{R}^{n+1}$, because $\{0\}$ is closed in the natural topology of $\mathbb{R}$ and $f$ is a continuous map (for the same topology).

Note that this question makes sense on $\mathbb{R}$ or $\mathbb{C}$, but not on an arbitrary field $\mathbb{F}$ since in general you do not have a topology on $\mathbb{F}^{n+1}$.