Thanks @PeterDonis. I think @PAllen makes good points. I would only make one additional point (with a degree of sophistry, it must be said) which is that the crossing point of the time-like curves doesn't exist in a realistic black hole. I think I see the point about there needing to be spacelike geodesics in my spacelike volume to be able to call it "space" rather than a space-like foliation. If the natural concept of "going straight ahead not advancing in time" takes you out of the volume then it's a bit difficult to argue that the volume represents a "slice of spacetime at a single time". Would you argue that the volumes in Minkowski spacetime defined by equal Rindler time aren't "space" on this basis?