Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the space of all compact subspaces of $R^\infty$ with the Hausdorff metric. What is known about this space? I vaguely recall that it is related to Waldhausen's $A$-theory - are there any references for that?

Motivation: if one instead looks at the space of all subspaces homeomorphic to or respectively diffeomorphic (if $X$ is a manifold) to $X$ one has a model for $BHomeo(X)$ or $BDiff(X)$ (since the corresponding space of embeddings of $X$ in $R^\infty$ is weakly contractible and this space is the quotient of the space of embeddings by the homeo/diffeo group). I have been having fun lately thinking about these models when $X$ is a finite set of points or a surface. This space of subspaces homotopy equivalent to $X$ is different - in particular I do not see it (directly anyways) as a model for $BHomotopySelfEquiv(X)$ - so I'd like to know more what is known about it.