Clearly, the set $\{x: P(x) \geq 0 \}$ consists of a finite union of intervals for the polynomial function
$P(x) = \sum_{n=1}^N a_n x^n$ with $a_n \in \mathbb{R}$.
Is there a similar result for a polynomial in several variables, i.e., a function $P: \mathbb{R}^k \to \mathbb{R}$ with
$P(x_1,\ldots, x_k) = \sum_{n=1}^N a_n x_1^{b_n^1}\cdots x_k^{b_n^k}$ with $a_n \in \mathbb{R}$
In particular, I would be interested in a result that states that
$A = \{ x: P(x) \geq 0\} = \bigcup_{i \in I} A_i$
where the $A_i$ are connected, closed sets in $\mathbb{R}^k$ with the cardinality of $I$ being at most countable (preferably finite).
I see that the regions must be connected and closed (by continuity) and they are separated from $A^c$ by the curve $\{x \in \mathbb{R}^k: P(x) = 0 \}$, but I am somewhat lost on the issue of the cardinality of $I$.