\(z\)-embedding and fibered product (Q1355416)

The pullback of \(f:X\to Z\) and \(g:Y\to Z\) here is called fibered product of \(X\) and \(Y\) with respect to \(Z\), \(f\) and \(g\). For a real function \(f:X\to \mathbb{R}\), the set \(f^{-1}(0)\) is the zero-set of \(f\). A subspace \(S\) of a space \(X\) is \(z\)-embedded in \(X\) if every zero set of \(S\) is the intersection of \(S\) and a zero set of \(X\). Passing to the Stone-Čech compactifications of involved spaces, and the Stone-Čech extensions of involved maps, one has the pullback of \(\beta f:\beta X\to\beta Z\) and \(\beta g:\beta Y\to \beta Z\), which contains the pullback of \(f:X\to Z\) and \(g: Y\to Z\). Is this inclusion \(z\)-embedding? Three equivalent statements to the statement that the mentioned inclusion is a \(z\)-embedding are found.