# 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

• 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 Tamaroff Mar 2 '15 at 22:11