  MacPherson conjectured that the Grassmannian $\mathrm{Gr}(2, \mathbb{R}^n)$
has the same homeomorphism type as the combinatorial Grassmannian
$\|\mbox{MacP}(2,n)\|$, while Babson proved that the spaces
$\mathrm{Gr}(2,\mathbb{R}^n)$ and $\mathrm{Gr}(1,2,\mathbb{R}^n)$ are homotopy
equivalent to their combinatorial analogs $\|\mathrm{MacP}(2,n)\|$ and
$\|\mbox{MacP}(1,2,n)\|$ respectively. We will prove that $\mathrm{Gr}(2,
\mathbb{R}^n)$ and $\mathrm{Gr}(1,2, \mathbb{R}^n)$ are homeomorphic to
$\|\mbox{MacP}(2,n)\|$ and $\|\mbox{MacP}(1,2,n)\|$ respectively.
