I am unaware of what the formal definition of "limit" is for a sequence of flats, but for the purpose of this question her is the definition of limit that I am using: Consider a sequence $s_1, s_2, s_3,$ etc. where $s_i$ is a flat of $R^m$ for all $i$. We will say that $lim_{n\to\infty}s_n=S$ if and only if for any point $p$ on $S$ and any positive value $\epsilon$, there exists a sufficiently large number $b$ such that if $x>= b$, the minimum distance between $p$ and $s_x$ will be no greater than $\epsilon$.

Here is my question:

1) Does there exist a homomorphism between the set of all n-flats in $R^m$ to some manifold, for all n and m, and 2) If so, to what manifold?

This question is partially motivated by a different question I asked earlier, where it was established that the set of all lines in a plane could not be continuously and bijectively represented by the set of all points in the plane, because the set of lines in a plane is homomorphic to an open mobius strip. Continuous bijective way of representing a line on a plane [answered]

Note: I know very little topology, so I would appreciate it if people provided links if they refer to even "well known" theories.