Example of Non-Compact Closed Set contained in Open Set With Special Property

Question

I am looking for an example of an open set $A$ in a metric space $X$ and a closed, non-compact subset $B$ of $A$ such that there is no $\delta > 0$ s.t. $\{x: \textrm{dist}(x,B) < \delta\} \subset A$.

I have tried using the discrete metric and other obvious tricks but have been unable to come up with an example. Any hints would be appreciated.

Answer 1

Let $X= \Bbb R^2.$ Let $A=\{(x, y)~|~ y \gt 0 \}.$ Let $B=\{(x, y)~|x \gt 0 \land xy \geq 1 \}.$

Choose $\delta \gt 0$. Then $(2\delta, \frac{1}{2\delta}) \in B$, so $(2\delta, -\frac{1}{4\delta}) \notin A$ but $\operatorname{dist}((2\delta, -\frac{1}{4\delta}), B) \leq \frac{3\delta}{4} \lt \delta.$