# Uniform Continuity in Topological Spaces?

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.

• This is almost the definition in term of uniformities ( entourage form). – Henno Brandsma Dec 9 '18 at 16:31
• Look up uniform spaces. – Lee Mosher Dec 9 '18 at 16:50

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