  We study $\textrm{Sym}(\infty)$-orbit closures of not necessarily closed
points in the Zariski spectrum of the infinite polynomial ring
$\mathbb{C}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. Among others, we characterize
invariant prime ideals in this ring. Furthermore, we study projections of basic
equivariant semi-algebraic sets defined by $\textrm{Sym}(\infty)$ orbits of
polynomials in $\mathbb{R}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. For $n=1$ we
prove a quantifier elimination type result which fails for $n>1$.
