I am trying to do a problem in Arbib's Category Theory book. Loosely rephrased:
Let $\{(X_i,\tau_i)\}_I$ be a family of topological spaces, $X$ a set, and $\{f_i:X\to X_i\}_I$ a family of functions. Define:
$$\tilde A:=\bigcup_{i\in I} \Big\{f^{-1}_i(A):A\in\tau_i\Big\}\subseteq \mathcal{P}\,(X)$$
and define $\tau$ as the intersection of all topologies on $X$ that contain $\tilde A$.
Then show that $\tau$ is the only topology on $X$ that makes the family of the $f_i$ optimal. I.e., given $(Y,\sigma)$ topological space and $g:Y\to X$ such that for all $i\in I$, $f_i \circ g$ is continuous, then $g$ must be continuous.
I have proved almost everything, including that if $A\in \tilde A$ then $g^{-1}(A)\in\sigma$. However, I believe it's possible to have open sets in $\tau\setminus \tilde A$. In this case, I do not know how to prove that their pre-image by $g$ is open in $Y$.
Obviously this needs to use the fact that $\tau$ is somehow generated by $\tilde A$. But I have no guarantee of $\tilde A$ being a basis or even a sub basis for the topology (no idea if their union equals $X$).
Any help or tip or insight would be appreciated. My hope is to be able to construct an arbitrary element of $\tau\setminus \tilde A$ from elements in $\tilde A$.
It's useful to know that in order to show that a function $g:Y\to X$ is continuous, it suffices to show that for each $S\in\cal S$, where $\cal S$ is a subbase, the preimage $g^{-1}(S)$ is open. Remember that by taking all finite intersections of element in $\cal S$ we obtain a base $\cal B$ for the topology on $X$, and each open set is then an arbitrary union of elements in $\cal B$. So if $U$ is open in $X$, $g^{-1}(U)$ is a union of finite intersections of elements of the form $g^{-1}(S)$, where $S\in\cal S$, hence the preimage of $U$ is open.
The smallest topology containing $\tilde A$ is the topology having $\tilde A$ as a subbase, so if you have shown that $g^{-1}(A)$ is open for each $A\in\tilde A$, then you are finished.