general relativity - Does any spacetime admit a global foliation in spacelike hypersurfaces?

In the comments of this question the following new questions came up: in general relativity, local coordinates can be found around any point, that single out a time coordinate and a three dimensional spatial submanifold, but in general it won't be possible to write spacetime globally as a product of a spacelike and a timelike submanifold.

If local coordinates around all points such that the locally spatial hyperplanes map into hyperplanes under a change of coordinates, then we got a foliation of spacetime into spacelike hypersurfaces.

I would be interested to know:

1. Does there always exist a global foliation of spacetime into spacelike hypersurfaces?


2. In such a foliation, are all leaves homeomorphic?