The topology of space is usually assumed simply connected, but could be multi-connected. We review in the latter case the topological constraints for all defects arising at high energy phase transitions and the possibility that the extended ones might still be present. We find that this is very unlikely. We conclude that observing a cosmic string or a domain wall now implies space to be effectively simply connected on scales up to the horizon size. (C) 1997 Elsevier Science B.V.
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