The definition of the weak topology(projective topology) in Kelley-Namioka 's book In Kelley-Namioka "Linear Topological Spaces",Page 31, it gives two equivalent definitions of projective topology(i.e.,weak topology). I don't know why they are equivalent. 
The point is I don't kown why the following statement is equivalent to the definition of the weak topology:
"The projective topology(A.K.A. the weak opology) can be described alternatively by specifying that a subset $U$ of $X$ is open relative to te projective topolgy if and only if $U$ is the inverse under $P$ of an open subset of $\prod_{f\in F}{Y_f}$, where $P$ is the map which sends a point $x$ of $X$ into the point with $f$-th coordinate $f(x)$(that is, $P(x)(f)=f(x)$)"
I hope someone can tell me how to prove it.
 A: Suppose that $F$ is a family of maps from a set $X$ to spaces $Y_f$ indexed by $F$. Let $Y=\prod_{f\in F}Y_f$. Let $$P:X\to Y:x\mapsto\langle f(x):f\in F\rangle$$ be the evaluation map. One way to show that the weakest topology making each $f\in F$ continuous is the same as the weakest topology making $P$ continuous is to show that a topology $\tau$ on $X$ makes $P$ continuous iff it makes each $f\in F$ continuous.
Suppose that $\tau$ is a topology on $X$ making each of the maps $f$ continuous, so that for each $f\in F$ and each open set $U\subseteq Y_f$, $f^{-1}[U]\in\tau$. Then $U$ is the union of basic open sets of the form
$$B=\{y\in Y:\forall f\in F_0(y_f\in U_f)\}\;,\tag{1}$$
where $F_0$ is a finite subset of $F$, and $U_f$ is an open set in $Y_f$ for each $f\in F_0$. Thus, to show that $P^{-1}[U]\in\tau$, it suffices to show that $P^{-1}[B]\in\tau$ for each $B$ of the form $(1)$. But 
$$\begin{align*}x\in P^{-1}[B]&\text{ iff }P(x)\in B\\
&\text{ iff }P(x)_f\in U_f\text{ for each }f\in F_0\\
&\text{ iff }f(x)\in U_f\text{ for each }f\in F_0\\
&\text{ iff }x\in\bigcap_{f\in F_0}f^{-1}[U_f]\;,
\end{align*}$$
so $$P^{-1}[B]=\bigcap_{f\in F_0}f^{-1}[U_f]\;.$$
By hypothesis each of the sets $f^{-1}[U_f]\in\tau$, so $P^{-1}[B]\in\tau$ as well, and so is any union of such sets.
Conversely, suppose that $\tau$ makes $P$ continuous. Then $P^{-1}[B]\in\tau$ for every basic open set in $Y$ of the form $(1)$. In particular, for each $f\in F$ and any open set $U_f\subseteq Y_f$, $B=\{y\in Y:y_f\in U_i\}$ is open in $Y$, and therefore $P^{-1}[B]\in\tau$. But 
$$\begin{align*}
P^{-1}[B]&=\{x\in X:P(x)\in B\}\\
&=\{x\in X:P(x)_f\in U_f\}\\
&=\{x\in X:f(x)\in U_f\}\\
&=f^{-1}[U_f]\;,
\end{align*}$$
so $f^{-1}[U_f]=P^{-1}[B]\in\tau$.
This shows that the topologies making $P$ continuous are exactly the same as the topologies making every map in $F$ continuous, and it follows immediately that the weakest topology making $P$ continuous is the weakest topology making every map in $F$ continuous.
