Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?

Question

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 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 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.

Answer 1

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.