Let $X$ be a separable Banach space and $D\subseteq X$ be a 


* proper, connected, and dense $G\_{\delta}$ subset of $X$,
* $X-D$ is $\sigma$-porous.


*Then is $X-D$ contained in a finite-dimensional subspace $E$ of $X$?* 


\*\*This seems at least plausible since Corollary [3.4 of this paper](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-38.1.125) shows that the codimension of a prous set in (finite-dimensions) is less than the entire space... \*\*


**Background/Definition(s)**


* *σ-porous:* A set $A$ is $\sigma$-porous if it can be covered by a *countable* number of porous sets.
* *porous:* A set $A$ is [porous](https://en.wikipedia.org/wiki/Porous_set) if for each $a \in A$, every $\lambda\in (0,1)$, and every $\epsilon>0$, there exists some $b \in X-A$ satisfying
$$
d(a,b)< ε \mbox{ and } B(b,\lambda d(a,b)) \cap A=\emptyset.
$$
* Facts: I can be shown that if $X=\mathbb{R}^n$ then any porous set is of Lebesgue measure $0$, Haar-null, and nowhere dense. However, it can be shown that (even in this finite-dimensional setting) there exist sets which are either Lebesgue measure $0$, Haar-null, or nowhere dense but fail to be $\sigma$-porous.



Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X=\ell\_\infty$ and let $D$ be the complement of the set $\{(x\_n)\in\ell\_\infty:x\_n\neq 0\text{ for at most one $n$}\}$. This last set is clearly closed, so $D$ is open, in particular $G\_\delta$. It is further clearly a proper, connected, dense subset. Let us now show that $X-D$ is porous, hence $\sigma$-porous.


Let us take $\alpha=1/4$ and arbitrary $x\in X,r>0$. The ball $B(x,r/2)$ necessarily contains a point $y$ all of whose coordinates have absolute value at least $r/4$. Now the ball $B(y,r/4)$ is contained in $B(x,r)$ and is clearly disjoint from $X-D$. Thus we conclude $X-D$ is porous.


Finally, $X-D$ is not contained in a subspace of finite dimension, since $(1,0,0,\dots),(0,1,0,\dots),(0,0,1,\dots)$ is an infinite linearly independent subset of $X-D$.


In fact, let me note that almost the exact same construction shows that your claim regarding Corollary 3.4 of the paper is *incorrect*, at least when understood in terms of the dimension of the subspace spanned. The only claims the paper makes are about the Hausdorff dimension of that set.


