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Is it already in literature this generalized notion of uniform continuity in an arbitrary topological space (not necessarily in exactly the same form)? Let $(X, T_{1})$, $(Y, T_{2})$ be topological spaces; let $f: X \to Y$. Then $f$ is called uniformly continuous-($T_{1}, T_{2}$) if and only if for every $U \in T_{2}$ there is some $V \in T_{1}$ such that $x,y \in V$ imply $f(x), f(y) \in U$.

In particular, if the codomain is restricted to be $\mathbb{R}$ with the standard topology, is the corresponding definition existing in literature?

Thanks.

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    $\begingroup$ This is almost the definition in term of uniformities ( entourage form). $\endgroup$ – Henno Brandsma Dec 9 '18 at 16:31
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    $\begingroup$ Look up uniform spaces. $\endgroup$ – Lee Mosher Dec 9 '18 at 16:50
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The definition you've proposed is a weaker (instead of stronger) form of classical continuity. Indeed, if $f$ is continuous then for an open $U$ put $V:=f^{-1}(U)$ to satisfy your condition.

Now not every function that satisfies your property is continuous. Take

$$f:\mathbb{R}\to\mathbb{R}$$ $$f(x)=\begin{cases}x &\text{if }x \neq 0 \\ 1 &\text{otherwise} \end{cases}$$

With the standard topology on $\mathbb{R}$ on both sides we have that $f$ satisfies your condition but is not continuous. The point is that we can always choose $V$ to avoid discontinuity at $0$.

There's no way to define uniform continuity on a general topological space. For that you need some additional structure.

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