Recall that a path in a topological space $X$ is a continuous map $[0,1] \rightarrow X$, where $[0,1]$ is the real unit interval.
Now, thinking about paths, and also about the $\pi_1(X,x)$, I can only imagine paths which really looks like some kind of "curved segments", which naively looks like "a segment", perhaps very strange, but segment-like.
There is a topology, for every set $X$, for which there are continuous maps from the unit interval which are not like that naive idea I have of a loop, the trivial topology. The problem is, that in the trivial topology the points of $X$ are not distinguishable, and so I find that this particular example is as trivial as the path $[0,1] \rightarrow \{pt\}$ (where $pt$ stands for point), and so it is not a useful example, for me, in a topological and geometrical sense.
So, I'm asking:
- Are there examples of paths in a space $X$ which are different from what a path looks like in a (say) $CW$-complex? (i.e. a curved, perhaps very strange, segment)
A path can be very unlike a curved segment for it can cross itself numerous times, even infinitely many times. An arc, which is a path with an image homeomorphic to [0,1] is exactly like a curved segment. In a Hausdorff space a path from a to b can be reduced or short cutted to an arc from a to b.