Hi.

Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, locally noetherian reduced or without embedded component schemes). Let $g:T\rightarrow S$ be a flat morphism with $T$ reduced or without embedded component.

Question: It is true that the fiber product $X_{T}:=X\times_{S} T$ is reduced or without embedded component?

I think that is true essentially because that the projection $X_{T}\rightarrow X$ is flat and then contract an irreducible component of $X_{T}$ on an irreducible component of $X$.

But universally open map have this propertie and reduced or without embedded component are not preserved by universally open base change....

Thank you.