In practical modelling, I'd argue that one of the most powerful concepts we have is the ability to interpolate functions between given data points. I want to ask, if any similar idea exists between topological spaces.
Suppose I have topological spaces $X$ and $Y$, and I want it that, at points $x_1,x_2...,x_n \in Y$ a map assumes values $y_1,y_2..,y_n \in Y$ would it be possible to interpolate this to a continous function between the spaces?