Product of a family of spaces of countable tightness

Question

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem?

Finite family of compact spaces of countable tightness has countable tightness.

Evey compact space has countable tightness?

Answer 1

A corollary of Exercise 3.12.8 (d) of Engelking's “General topology” claims: if $f:X\to Y$ is a closed continuous map of topological spaces, $Y$ is regular, and for each $x\in X$ we have $t(f(x),Y)\le\kappa$ and $t(x,f^{-1}(X))\le\kappa,$ then $t(x,X)\le\kappa$.

Not every compact space has a countable tightness, because $t(X)=w(X)$ for each dyadic compact $X$, by Exercise 3.12.12 (h) of the same book.