Let $K$ be a non-empty compact subset of $\mathbb{R}^2$ .
Is $U = \mathbb{R}^2 / K$ ever simply-connected?
Perhaps Alexander duality is relevant, but the relevant cohomology is a bit tricky to compute if $K$ is not locally contractible.
Unfortunately, some mental error has occurred between recall of the hypothesis of the statement and transcription of the question. $K$ is in fact not compact, but instead can be generated by finite unions and arbitrary intersection of sets of the form $A\times \mathbb{R}$ and $\mathbb{R}\times B$, with $A,B$ compact. Apologies.