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!
I don't think you can expect any condition like that. For any closed set $F$ with nonempty interior, most often you can write $F=E\cup S$ for disjoint dense sets $E,S$ (for example $F=[0,1]$, $E$ the rationals in the interval, $S$ the irrationals; or $F=C[0,1]$, $S$ the polynomials, $E=F\setminus S$). Then the sets satisfy your conditions but $\overline E=F$ will always have balls.
Note that in the second example mentioned above, $S$ is a subspace as in your problem.