From Wikipedia
let $A$ be a subset of $\mathbb{R}^n$. A function $f : A → \mathbb{R}^m$ is uniformly continuous if and only if for every pair of sequences $x_n$ and $y_n$ such that $ \lim_{n\to\infty} |x_n-y_n|=0\, $ we have $ \lim_{n\to\infty} |f(x_n)-f(y_n)|=0.\, $
I was wondering if this can be generalized to $f : X → Y$ when $X$ is a metric space and $Y$ is $\mathbb{R}^m$ or even another metric space? If "if and only if" doesn't hold, does "if" or "only if" hold?
Are there generalizations when $X$ is a uniform space and $Y$ is $\mathbb{R}^m$ or even another uniform space? For example, by replacing sequence with net or filter, and distance with entourage?
Thanks and regards!