Set operations in the constructions of the Weak Topology cannot be reversed Let $X$ be a generic set and let $(Y_i)_i$ be a family of topological spaces. Let $(\varphi_i)_i$ be a collection of functions of the kind $X \to Y_i$. It is possible to determine a topology (that would be the coarser) in which all those functions are continuous, as follows:
1) if $\omega_i$ is an open set from some of the $Y_i$, then $\varphi_i^{-1}(\omega_i)$ is necessarily an open set (because we are supposing the functions to be continuous). So we can obtain a family $U$ of subsets in $X$ given by these pre-images.
2) We can consider finite intersections of members from $U$ obtaining a space $\phi$ that includes $U$ and that is stable under finite intersections. $\phi$ may not be stable for arbitrary unions.
3) Then we can consider the family $\mathcal{F}$ obtained by forming arbitrary unions of elements from $\phi$. It can be proven that $\mathcal{F}$ is stable under arbitrary unions and finite intersections.
The process of taking finite intersections first and then arbitrary unions cannot be reversed because we can obtain a family of subsets that is not stable under arbitrary unions.

Do you know some concrete examples showing why the "reverse" construction fails? 

 A: There have to be at least two $Y_i$, since the preimages of open sets of a single $Y$ are already closed under arbitrary unions and finite intersections. We can construct an example using the family from Guiseppe's comment by taking $Y_1$ to be $\mathbb R$ endowed with the open sets $(-\infty,a)$ for $a\in[-\infty,\infty]$ and $Y_2$ to be $\mathbb R$ endowed with the open sets $(b,\infty)$ for $b\in[-\infty,\infty]$, with $\phi_i(x)=x$.
An example with standard topologies is $\mathbb R^2$ with $Y_1=Y_2=\mathbb R$, with $\phi_i$ the projection onto the $i$-th component. Then the standard topologies on $Y_1$ and $Y_2$ induce the standard topology on $\mathbb R$, since the subbase of preimages of open sets of $Y_1$ and $Y_2$ consists of the sets $U\times\mathbb R$ and $\mathbb R\times V$, with $U,V\subseteq\mathbb R$ an open set. Finite intersections yield sets of the form $U\times V$, and then arbitrary unions lead e.g. to
$$\bigcup_{t\in\mathbb R}\left((t,\infty)\times(-\infty,t)\right)\;,$$
the open set of points strictly under the main diagonal. This cannot be formed using a finite intersection of arbitrary unions, since each term of the intersection would have to cover the entire set, which implies that it must contain a set of the form $\mathbb R^2\setminus\left([-\infty,t]\times[t,\infty]\right)$, and each such set can only exclude one point of the diagonal from the intersection, so it would require (uncountably) infinitely many of them to exclude the entire diagonal.
A: Since it's tagged functional analysis and is linked with the weak topology, I will give an example in this spirit (hoping it's correct), although I won't consider all the linear maps. Take $X=\ell^{\infty}(\Bbb R)$, the space of bounded sequence of real numbers endowed with the uniform norm. Consider $f_n$ the (continuous) linear functional defined by $f_n(\{x_n\}_k)=x_n$, and put for $r$ real number:
$$O_r:=\bigcup_{n\in\Bbb N}\{x\mid x_{2n}<r\},\quad O'_r:=\bigcup_{n\in\Bbb N}\{x\mid x_{2n+1}<r\}.$$
Assume that we can write $\bigcup_{r<0}(O_r\cap O'_r)$ as an union of the form $\bigcup_{j\in J}f_j^{-1}(O_j)$, where $O_j$ is an non-empty open subset of the real line and $J\subset \Bbb N$. Assume that $2j\in J$ for some $j$ (otherwise take $2k+1$ for some $k$), and $r_{2j}\in O_{2j}$. The sequence $x=r_{2j}e_{2j}$ is in $f_{2j}^{-1}(O_{2j})$ but not in $\bigcup_{r<0}(O_r\cap O'_r)$. 
So after taking the arbitrary unions and finite intersections, we have to take again the arbitrary unions of such sets, which finally gives the same result.
