Speaking just about space, we say that the universe is either open (topologically $E^3$) or closed (topologically $S^3$). But since a metric connection defines curvature on spacetime and not just space, does this mean a closed universe (having positive curvature) is also a closed spacetime (topologically $S^4$)? In other words, would positive curvature with a closed universe imply that we are on a closed time-like curve? Otherwise, spacetime would be open (topologically $E^4$ or $S^3\times E^1$) even if space is closed.
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No. For example, in the Friedmann metric with $k=+1$, space is closed but timelike curves in spacetime are not closed.
