Let $X, Y$ be two topological spaces and let $\mathcal F$ be a family of functions from $X$ into $Y$. What is the definition of equicontinuity for $\mathcal F$?
If $Y$ was a metric space then $\mathcal F$ is equicontinuous at $x$ if for each $\epsilon>0$ there is a neighborhood $U$ of $x$ such that $|f(x)-f(y)|< \epsilon$ for all $y\in U$ and all $f\in \mathcal F$.
However I can not extend this definition to a topological space.
Not surprising, since this concept does not exist for topological spaces. However, there is a concept of a uniform space which is somewhere between a metric space and a topological space. (Every metric space is a uniform space and every uniform space is a topological space.) Equicontinuity still makes sense for uniform spaces, see here.