  For a smooth spacetime $X$, based on the timelike homotopy classes of its
timelike paths, we define a topology on $X$ that refines the Alexandrov
topology and always coincides with the manifold topology.
  The space of timelike or causal homotopy classes forms a semicategory or a
category, respectively. We show that either of these algebraic structures
encodes enough information to reconstruct the topology and conformal structure
of $X$. Furthermore, the space of timelike homotopy classes carries a natural
topology that we prove to be locally euclidean but, in general, not Hausdorff.
  The presented results do not require any causality conditions on $X$ and do
also hold under weaker regularity assumptions.

Differential Geometry