Given a path-connected topological space $X$ (lets say compactly generated; this entire post will be working in the category of compactly generated topological spaces) with a designated point $x$, we can form the loop space $\Omega X$ of pointed morphisms from $S^1$ to $X$. This has the natural structure of a topological group.
Now given a topological group $G$, we can form its classifying space $BG$. I believe that it is true that $X$ is homotopy equivalent to $B\Omega X$; is this true?
I suspect that you could form some model category of topological groups and some model category of connected topological spaces, and show that $\Omega$ and $B$ are a Quillen equivalence. For one thing, I know that $\Omega BG$ is homotopy equivalent to $G$, and I have a feeling that if I could write out the model structures this should become clear. On the other hand I'm not that fluent in model category theory, and I'm also wondering if there's an elementary way to see this.