  It is known that a limit $(M^n_j,g_j)\to (X^k,d)$ of manifolds $M_j$ with
uniform lower bounds on Ricci curvature must be $k$-rectifiable for some unique
$\dim X:= k\leq n = \dim M_j$. It is also known that if $k=n$, then $X^n$ is a
topological manifold on an open dense subset, and it has been an open question
as to whether this holds for $k<n$.
  Consider now any smooth complete $4$-manifold $(X^4,h)$ with
$\text{Ric}>\lambda$ and $\lambda\in \mathbb{R}$. Then for each $\epsilon>0$ we
construct a complete $4$-rectifiable metric space $(X^4_\epsilon,d_\epsilon)$
with $d_{GH}(X^4_\epsilon,X^4)<\epsilon$ such that the following hold. First,
$X^4_\epsilon$ is a limit space $(M^6_j,g_j)\to X^4_\epsilon$ where $M^6_j$ are
smooth manifolds with $\text{Ric}_j>\lambda$ satisfying the same lower Ricci
bound. Additionally, $X^4_\epsilon$ has no open subset which is topologically a
manifold. Indeed, for any open $U\subseteq X^4_\epsilon$ we have that the
second homology $H_2(U)$ is infinitely generated. Topologically, $X^4_\epsilon$
is the connect sum of $X^4$ with an infinite number of densely spaced copies of
$\mathbb{C} P^2$ .
  In this way we see that every $4$-manifold $X^4$ may be approximated
arbitrarily closely by $4$-dimensional limit spaces $X^4_\epsilon$ which are
nowhere manifolds. We will see there is an, as now imprecise, sense in which
generically one should expect manifold structures to not exist on spaces with
higher dimensional Ricci curvature lower bounds.
