Given a topology $\tau$ for a set $S$, one can easily check if some functions are continuous between some subsets $A,B \subseteq S$, inheriting a topology from $\tau$ to both. My question is if/how I can do the other way around. Choosing the kind of function $f: A \rightarrow B$ that I want to be continuous between $(A, \tau|_A)$ and $(B, \tau|_B)$, how can i construct such $\tau$?
To help understand what I mean, I'll give an example. The continous deformations (not creating holes or glueing point) can be checked to be continuous under the standard topology in $\mathbb{R}$. But suppose I don't know the existence of the standard topology, can I deduce it just by deciding that I want the continuous deformations to be my continuous functions between topological spaces with such topology? If so, there is a "standard method" to deducing such topology?
Best Answer
For whatever map $f$, the trivial topology is a topology that will make $f$ continuous for any $A,B \subseteq S$. So I don't think that you can recover a specific topology. The trivial topology will always satisfy your requirement.