Selections, games and metrisability of manifolds
  In this note we relate some selection principles to metrisability and separability of a manifold. In particular we show that $\sf{S}_{fin}(\mathcal K,\mathcal O)$, $\sf{S}_{fin}(\Omega,\Omega)$ and $\sf{S}_{fin}(\Lambda,\Lambda)$ are each equivalent to metrisability for a manifold, while $\sf{S}_1(\mathcal D,\mathcal D)$ is equivalent to separability for a manifold. 
David Gauld
math.GN