E and S are subsets of a metric space. $E$ is a subset of $\bar{S}\backslash S$. Then $\overline{E}\subset(\overline{S}\backslash S^{o})$, but I wonder whether there is some condition that guarantees or forbids $\overline{E}$ to contain an open ball in $\overline{S}$.
For people interested in the background, I am looking at something in operator algebras. $S\subset\mathcal{L}(X)$ is a subspace of operators on a Banach space and I want to use the properties of operators in $S$ to get some result about $\overline{S}$.
However, the argument clashes if the exceptional set $E$ is not nowhere dense. So I wonder whether there is some condition on $S$ or $X$ or whatever under which we can eliminate this possibility.
The question is actually quite open. I think any condition on either the underlying space $X$, or the operators on $S$ would be of great help. Actually even a condition that would imply that $\overline{E}$ contains an open ball would lead to something interesting in the other direction.
Thanks!