Elementary topology problem. Let $ ((Y_{\alpha},\tau_{\alpha}) \mid \alpha \in J) $ be a $ J $-indexed family of topological spaces and $ X $ any non-empty set. Let $ (f_{\alpha} \mid \alpha \in J) $ be a $ J $-indexed family of functions, where $ f_{\alpha}: Y_{\alpha} \to X $. What topology $ \tau $ can you put on $ X $ that will make all of the $ f_{\alpha} $’s continuous with respect to the $ \tau_{\alpha} $’s and $ \tau $?
Please help me with this. I think that I only need to put the indiscrete topology on $ X $, so that the only open sets in $ X $ are $ X $ itself and $ \varnothing $, and the inverse image of each of these under $ f_{\alpha} $ is open in $ Y_{\alpha} $.

One can define a topology $ \tau $ on $ X $ as follows:

Declare a subset $ U $ of $ X $ to be $ \tau $-open if and only if $ {f_{\alpha}^{\leftarrow}}[U] \in \tau_{\alpha} $ for each $ \alpha \in J $.

Then $ \tau $ is the finest topology on $ X $ that makes $ f_{\alpha}: (Y_{\alpha},\tau_{\alpha}) \to (X,\tau) $ continuous for each $ \alpha \in J $.
