About the definition of universal covering space

Question

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

Answer 1

Most authors define a universal covering as a simply connected covering space of a path-connected locally path-connected space $X$.

Note that the requirement that $X$ be path-connected can be omitted. In fact, as the continuous image of a simply connected (in particular path-connected) space, $X$ itself must be path-connected.

However, we shall see later that it is not a good a idea to drop the requirement of local path-connectedness.

Let us understand what makes a universal covering space "universal". We consider a connected space $X$ and all covering spaces $p : \tilde X \to X$ with connected $\tilde X$. Such a covering space $p : \tilde X \to X$ is called universal if all other connected covering spaces $p' : \tilde X' \to X$ lie "below it" in the sense that there exists a map $f : \tilde X \to \tilde X'$ such that $p' \circ f = p$. See for example Greenberg & Harper p.27.

It is a well-known theorem that all simply connected covering spaces over a locally connected base are universal in that sense. This is why most authors define it as in the first sentence above. However, there are connected localy path-connected spaces which have a non-simply connected universal covering space in the sense of our general definition. An example is given in

Edwin H. Spanier, Algebraic topology (Springer Science & Business Media, 1989).

Why is it not good idea to drop the requirement of local path-connectedness? Most interesting theorems in the theory of covering spaces, e.g. the Lifting Criterion, hold true only under this assumption. Here is an example of a simply connected covering space over a non-locally path-connected space which is not universal in the above sense.

Take $W$ be the Warsaw circle which is a simply connected non-locally path-connected space. The identity $id_W$ would be a universal covering in the "Wikipedia sense". There are many other connected covering spaces $p : \tilde W \to W$ with more than one sheet. See my answers to Isomorphism between deck transformations and permutations onto the fiber and Classification of covering spaces for spaces that are not locally path connected: counterexamples?

However, there is no $f : W \to \tilde W$ with $p \circ f = id_W$. Such $f$ would be a section of $p$, and this is impossible. See When does a covering map have a section?

Answer 2

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.