restriction map in Stone space is open

Question

In the Stone space, how to proof that the restriction map $S_{m+n}(B)\to S_{n}(B)$ is open? Where B is subset of the model of the theory. I know that the restriction is continuous and surjective.

Answer 1

Hint: A basic open subset $U_\varphi$ of $S_{n+m}(B)$ consists of all $(m+n)$-types containing some formula $\varphi(x_1,\dots,x_{n+m})$. Prove that the image of $U_\varphi$ under the restriction map is exactly $U_\psi$, where $$\psi(x_1,\dots,x_n)=\exists x_{n+1}\dots\exists x_{n+m}\varphi(x_1,\dots,x_{n+m}).$$