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Suppose $F$ is a closed set in a Euclidean space, and for $\epsilon>0$, let $V_\varepsilon$ be the $\varepsilon-$neighborhood of $F$ i.e. the set of points $x$ having a distance less than $\varepsilon$ from $F$. Consider the following condition on $F$:

A) There is $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$, the complement of $V_\epsilon$ is connected.

Are there some topological conditions applied to $F$ that imply condition A? By topological condition I mean for example that $F$ is contractible, or...?

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    $\begingroup$ A remark on $V_\epsilon$: in general, they do not form a basis of open neighborhoods of $F$ unless, say, $F$ is compact. Under some conditions, Alexander duality should give some information. $\endgroup$
    – Z. M
    Commented Feb 22, 2022 at 22:10
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    $\begingroup$ Simple remark: contractibility is not enough, as $S^1$ with a point removed shows $\endgroup$
    – Alessandro Codenotti
    Commented Feb 22, 2022 at 22:25

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