We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[x_ij:i∈N,j∈[n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[x_ij:i∈N,j∈[n]]. For n=1 we prove a quantifier elimination type result which fails for n>1.