Q:

continuity on product topology

Question:
  Let $f: X\rightarrow Y$ and $g: X\rightarrow Z$
  be continuous, where X,Y Z are topological spaces.
Define 
  $h: X \rightarrow Y \times Z$
$x  \mapsto \left ( f\left ( x \right ),g\left ( x \right ) \right )=h\left ( x \right )$
  for all $x \in X$
Prove that h is continuous wrt the product topology on $Y \times X$.

Attempt:
Let $\jmath$ ,$\imath$  and $\Im$ be topologies on X, Y and Z, respectively.
f and g are continuous function so by definition of continuous function on a topological space,
we expect:
$\forall u \in \imath: f^{-1}\left ( u \right ) \in \jmath$
$\forall \bar{z} \in \Im: g^{-1}\left ( \bar{z} \right ) \in \jmath$
Clearly, what I do need to show is that the pre-image of the open set in the product topology $\imath  \times  \jmath$ is mapped to an open set in $\jmath$
Any help is appreciated.
Thanks in advance.

A:

What do open sets in $Y\times Z$ look like? They are of the form $U\times V$ where $U$ is open in $Y$ and $V$ is open in $Z$. Then you should try to show that
$$h^{-1}(U\times V)=f^{-1}(U)\cap g^{-1}(V)$$
which is open in $X$ by continuity of $f$ and $g$.

