Definition: A set $M$, also in a metric space, is said to be uniformly locally connected if and only if for every $\varepsilon > 0$ there exists $\delta>0$ such that any pair of points $x, y$ of $M$ with $d(x,y) < \delta$ lie together in a connected subset of $M$ of diameter less than $\varepsilon$.
I know that if a metric space is compact and locally connected then it is uniformly locally compact (See A compact locally connected metric space is "uniformly locally connected" ) if instead of compactness we have local compactness, the result is true?. Intuitively I think not. Could you show me an example? thanks