What topology should I put on $\mathbb{R}[x_1,...,x_n]_d$ to show that the set of polynomials which are positive on $\Bbb R^n$ is closed?

Question

I am asked to prove that the set $$ P^n_d = \{f \in \mathbb{R}[x_1,...,x_n]_d | f(\alpha) \geq 0 \, \forall \alpha \in \mathbb{R}^n \} \subseteq \mathbb{R}[x_1,...,x_n]_d $$ is closed, but the topology is not specified. I tried searching for common topologies on the ring of polynomials $\mathbb{R}[x_1,...,x_n]$ but haven't found any. I have learned that it is a topological vector space but I still dont know which topology this refers to. Here $\mathbb{R}[x_1,...,x_n]_d$ denotes the vector space of polynomials of degree exactly $d$ in $n$ variables.

Answer 1

Pick any basis of this finite-dimensional topological vector space and you get an identification with $\mathbb{R}^N$; port the topology from that. It doesn't depend on the choice of identification because $GL_N(\mathbb{R})$ acts on $\mathbb{R}^N$ via homeomorphisms.