A basic-closed semialgebraic set in $\mathbb{R}^{n}$ is defined as $$ M=\lbrace x \in \mathbb{R}^{n} \mid f_{1}(x) \geq 0, \ldots, f_{m}(x) \geq 0 \rbrace$$ for some polynomials $f_{i}$ in $n$ variables over $\mathbb{R}$. Find an example of two disjoint basic-closed semialgebraic sets such that the union is not basic-closed semialgebraic.
My thought: If $M$ $(n=2)$ contains an algebraic set which lies in the interior and on the boundary, then I can show that this set cannot be basic-closed (like $M=\{x\ge 0 \text{ or }y\ge 0\}$ and $L=\{y=0\}$). But this is not a disjoint union. If $U=\{x\le −1 \text{ or }(y\ge 0 \text{ and } x\ge 0)\}$, how can I show that $U$ is not basic-closed?
The example is wrong; $U$ is basic-closed. But $W=\lbrace (x,y) \in \mathbb{R}^{2} \mid (x\geq 1, y \geq 1) \text{ or } x+y \leq 0 \rbrace $ should be better.