Uniform limit theorem for arbitrary uniform space.

Question

Definition: A function $f:X\to Y$ between uniform spaces is called uniformly continuous if for every entourage $V$ in $Y$ there exists an entourage $U$ in $X$ such that for every $(x_1,x_2)$ in $U$ we have $(f(x_{1}),f(x_{2}))$ in $V$.

I am trying to prove uniform convergence theorem in the most general setting. (without any norm, metric or pseudometric)

Assume we have a net $$(f_\alpha)_{\alpha\in I}$$

in a set of functions $$(E\rightarrow X)$$, such that each $$f_\alpha:E\rightarrow X$$ is continuous. Also it is given that $$f:E\rightarrow X$$ and $$(f_\alpha)_{\alpha\in I} \overset{unif}{\longrightarrow} f$$.

I want to show that f is continuous, but I stop on the first step of the proof: assume we have U an open subset of X. Somehow we need to show that $f^{-1}[U]$ is open.

1. Maybe I need additional requirements about how uniform structure on X connected with topology. Any ideas how to continue the proof and which properties I should require to make it happen?

2. Does a uniform structure uniquely determine topology on the set?

3. If continuous wrt this topology, would it be automatically uniformly continuous, or not?