Schematically, I understand the path space fibration $PX$ over some path-connected, pointed topological space $X$ with base point $x_o$ as: $$\Omega X \hookrightarrow PX \twoheadrightarrow X,$$ where the first arrow is the inclusion and the second is the evaluation map.
If we understand $PX$ as the space of paths in $X$ (i.e., continuous maps $p(t)$ from the unit interval to $X$, with $p(0) = x_o$), then it seems that the natural definition of $\Omega X$ is the space of paths $p(t)$ with $p(0) = p(1) = x_o$. However, I have found conflicting statements from various sources:
- This MSE question says that $\Omega X$ is the space I have just described.
- The Wikipedia article for path space fibration says that $\Omega X$ is the loop space of $X$; the Wikipedia article for loop space defines that space as the space of continuous pointed maps from $S^1$ (with base point) to $X$.
- This MSE question points out that the space of continuous pointed maps from $S^1$ to $X$ is not the same as the space of paths in $X$ beginning and ending at the same point, which I think is correct due to the fact that elements of the latter may be discontinuous at $x_o$ between $t=1$ and $t=0$.
Having become thoroughly confused by everything I found on the internet, I turned to Hatcher, where I discovered a lengthy and fairly dense discussion on loop spaces which requires the use of "James reduced products," a concept I have not encountered before in my coursework. This leads me to believe that perhaps the whole situation is more complicated than I initially thought.
My question is, what is $\Omega X$, and what subtlety am I missing here that is leading to the confusion described above?