Alternative definition of simply connected spaces by Hochschild I am reading Hochschild´s book "The Structure of Lie Groups" and in the section 4 about coverings he defines a space $P$ is simply connected if it satifies the following conditions:
(1) $P$ is connected and locally connected
(2) if $f: S \to T$ is a covering, $g$ a continuous map of $P$ into $T$, $p_0$ a point of $P$, and $s_0$ a point of $S$ such that $g(p_0)=f(s_0)$ then there is one and only one continuous map $h$ of $P$ into $S$ such that $g=f \circ h$ and $h(p_0)=s_0$.
On the other hand, he defines a space $P$ is arcwise simply connected as always a space is defined as simply connected, it is if $P$ is connected and locally path-connected, and if every closed arc in $P$ can be shrunk to a point.
I have thought about why he does that definition of simply connected and I hope maybe someone can enlighten me. For now I have found an example that is simply connected but are not arcwise simply connected. This is the The ordered square, which is Hausdorff, connected and locally connected, and it isn´t locally path-connected. However, what I really would like to find it is an example that is a topology group also because that is the context of the book. I suppose that this space must not satisfy the hypothesis of being locally path-connected so that by homogeneity no point has a base of local neighborhoods that are simply path-connected. The other example I may have it is $\mathbb{R}$ with the cofinite topology, which is connected and locally connected, it isn´t locally path-connected and Hausdorff, but I haven´t figured out if it is simply connected.
PS: I apologize for my English, I hope you can easily understand what I wrote.
 A: Hochschild's approach is highly unusual. The only explanation for his strange definition of "simply connected" is that he wanted to restrict his presentation of covering space theory to the absolute minimum needed in the rest of the book. Thus he avoided to introduce the fundamental group which is an essential ingredient in every standard treatise of covering space theory.
It is noteworthy that he defines his concepts of "simply" connected" and " arcwise simply connected" only for special classes of spaces (for connected locally connected spaces and connected locally arcwise connected spaces, repectively). This certainly has only technical reasons: (1) His concept of covering space is only defined in the realm of connected locally connected spaces (which makes quite sense). (2) In Theorem 2.1 he wants to prove that "arcwise simply connected" implies "simply connected", and this requires some technical prerequisites.
As you observed, "simply connected" is usually defined as arcwise (= pathwise) connected plus shrinkability of all closed arcs. Clearly connected locally arcwise connected spaces are arcwise connected, so Hochschild's definition of "arcwise simply connected" is nothing else than the standard definition of "simply connected" restricted to a special class of spaces.
In Theorem 2.1 he shows then that arcwise simply connected spaces have the usual lifting property, which is normally proved in a broader context using fundamental groups. See the section Lifting properties here.
In my opinion he should at least have mentioned that his notation is non-standard and should have encouraged the reader to study other books on covering maps.
I also recommend to read the review in zbMath:


