On vector lattices of continuous functions in locally compact spaces (Q1824821)

Two results on lattices of continuous functions are obtained which are connected with Gelfand-Kolmogorov and Stone-Weierstrass' theorems. First, it is shown that if, under natural assumptions, there exists a linear lattice isomorphism \(\phi\) : \(V_ 1(X)\to V_ 2(Y)\) of completely regular vector lattices over Tychonoff spaces X, Y, then X and Y are homeomorphic. Second, if X is a locally compact Hausdorff space, then every completely regular vector lattice over X is uniformly dense in the space \(C_ c(X)\) of all continuous real-valued functions on X having a compact support.