In big bang cosmologies with an initial singularity, spacetime only exists at $t>0$. Spacetime manifolds don't actually have singular points, they just have open edges beyond which they can't be extended. I'm not sure that the spatial topology at $t=0$ is a well defined concept, but the horizon problem suggests that it shouldn't be viewed as a single point in general. Cosmologies with positive spatial curvature can have a horizon problem, so even a finite total spatial volume that goes to zero as $t\to 0$ isn't enough to guarantee that the initial singularity can be reasonably treated as a point.
In cosmologies (probably based on quantum gravity) that really do start at a nonsingular point, there is no obvious obstacle to having noncontractible spatial slices at later times. Take the Euclidean plane with cosmological time being distance from the origin. Space is a point at $t=0$ and a circle at all $t>0$. This example has the wrong metric signature, but there is no obvious reason why a real quantum-gravitational cosmology couldn't have broadly similar features.