Hi guys,

The question I'm doing states,

"Let $\displaystyle i_1, i_2, \ldots, i_n$ be indices in $\displaystyle I$, and assume that for each $\displaystyle i \in I$, we are given $\displaystyle U_i \subseteq X_i$. Show that $\displaystyle \bigcap_{1\leq r \leq n} \pi^{-1}_{i_r}(U_{i_r})$ can be described as a product of subsets of $\displaystyle X_i$'s"

I just want to see if my thinking here is correct.

We have that $\displaystyle \pi^{-1}_{i} $ is given by the projection onto the ith factor. So, $\displaystyle \pi^{-1}_{i_1} : U_{i_1} \times U_{i_2} \times \cdots \times U_{i_n} \rightarrow U_{i_1}$, for instance. We have the following fact for some $\displaystyle U_1\subseteq X_1, U_2 \subseteq X_2$,

$\displaystyle \pi^{-1}_1(U_1) = U_1 \times X_2$

where $\displaystyle \pi_1 : U_1 \times U_2 \rightarrow U_1$

Now, I know I have to use this to prove the above question, I'm just stuck on how I would represent $\displaystyle \pi^{-1}_{i_r}(U_{i_r})$ using the above, and how to then manipulate the intersections. There is a further property that

$\displaystyle \pi^{-1}_1(U_1) \cap \pi^{-1}_1(U_2) = U_1 \times U_2$,

which perhaps may come in useful, although I haven't seen any use for it.

Thanks in advance,

HTale.