The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
A (possibly empty) family of morphisms $f_i:(X,\tau)\rightarrow (X_i, \tau_i)$ in $\textbf{Top}$ is $\textit{optimal}$ if whenever $g:(Y,\sigma)\rightarrow X$ (i.e., $(Y,\sigma)$ is a topological space and $g:Y\rightarrow X$ is a function) is such that $f_ig:(Y,\sigma)\rightarrow (X_i,Y_i)$ is continuous for all $i$ then $g:(Y,\sigma)\rightarrow (X,\tau)$ is continuous.
Given any set $X$ and functions prove that the intersection of all topologies containing $\cup\{f^{-1}_i(A)\mid A\in \tau_i\}$ is the unique topology $\tau$ on $X$ such that $f_i:(X,\tau)\rightarrow (X_i, \tau_i)$ is optimal. $\tau$ is called the $\textit{optimal lift of}$ $f_i:X\rightarrow (X_i, \tau_i).$ The reader familiar with the 'subspace topology' on a subset $A$ of a topological space $(X,\tau)$ should observe that this is just the optimal life of the inclusion map of $A$. Also, the empty case is the 'discrete topology' on $X.$
I have a few $\textbf{questions}$ about the exercises that are confusing me.
(1) I would like to know what are the relation between $X$ and $\tau$ to $X_i$ and $\tau_i$ respectively.
(2) for the maps $g:(Y,\sigma)\rightarrow X$, $f_i:X\rightarrow (X_i, \tau_i).$ I am not sure if I am understanding them correctly. For the case of $g:(Y,\sigma)\rightarrow X$, is $g$ defined as $g(O)=a$, where $O$ is an open set in $Y$ and $a$ is an element in $X$, similarly for $g:(Y,\sigma)\rightarrow X$, again is $f_i$ defined as $f_i(b)=W$, where $W$ is an open set in $X_i$ and $b$ is an element of $X$? (I am assuming that $(X,\tau)$ and $(Y,\sigma)$ are both topological spaces with $X$ and $Y$ being sets)
(3) Lastly, in the part where it asks the reader to prove, where it says "the intersection of all topologies containing $\cup\{f^{-1}_i(A)\mid A\in \tau_i\}$ is the unique topology $\tau$ on $X$...". In mathematical notation, does it translate to the following: let $\{T_i\}_i\in I$ be a family of topologies on set $X$ such that $\cap T_i=\tau,$ and $\cup\{f^{-1}_i(A)\mid A\in \tau_i\}\subset \cap T_i?$
Thank you in advance