Let $B$ be a topological space. A universal covering of $B$ is defined to be a simply connected covering space. Suppose $B$ has a universal covering. Is such a covering unique (up to a homeomorphism)?

When $B$ is locally path-connected, the lifting criterion easily implies the uniqueness of a universal covering. This can also be shown 'by hand' using the homotopy lifting property, or can be shown by playing with the fibered product of two universal coverings. But now $B$ is not assumed to be locally path-connected, and none of these work.

Question: Is the universal covering unique?

  • $\begingroup$ You don't need $B$ to be locally path connected. Universal coverings are homeomorphic by the universal property: en.wikipedia.org/wiki/Covering_space#Universal_covers $\endgroup$ – freakish Mar 10 '18 at 12:55
  • $\begingroup$ @freakish Do you have a proof for the universal property? $\endgroup$ – abccsss Mar 10 '18 at 13:27

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