About the definition of universal covering space

There are some references (for instance in Greenberg & Harper) that consider the universal covering space to be not only simply connected but also locally path connected. This definition seems to me a good one since the so called Lifting Criterion (c.f. Theorem 6.1, Greenberg & Harper), as it is proved in the latter reference, requires the condition of being locally path connected.

However, in most of definitions i see in other places (e.g. online notes, wikipedia, etc...) the universal covering space is only assumed to be simply connected. In this case, i don't know if that Lifting Criterion is still valid and thus, I'm not able to check the universal property of the universal covering space.

What is the most standard definition ? Thanks

1 Answer

As I understand it, the universal covering space is usually just defined to be a simply connected covering space. However, the fact that the space is simply connected already implies that it is locally path-connected, since a simply connected space is assumed to be path-connected.

However, if the base space is locally path-connected, then the covering space inherits this property since the covering map is a local homeomorphism. Also, it looks like a universal covering can only exist for spaces that are connected, locally path-connected and semi-locally simply connected, but this fact is beyond my current level of understanding.

• Path connected doesn't imply locally path connected. Consider say $(\Bbb Q\cap (0,1))\times [0,1]\cup [0,1]\times \{0,1\}$. (Assuming locally path-connected means having a basis of path connected neighborhoods at each point, which is more restrictive than the possible definition that every point has a path connected nbhd.)
– Pedro
Mar 2, 2015 at 22:11