Graphic interpretation of path fibration.

Question

Let $S^2$ the unit sphere. We can consider the associated path fibration $$ \Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a picture. Do you have some idea in order to do a picture (over $S^2$) of path fibration? In particular is there a way to draw the open sets in $P(S^2)$ with the compact open topology?

Answer 1

I would suggest taking some compact subset $K$ in $I$ and some open subset in $S^2$ and then drawing examples of single paths which would belong to the subbase set $V(K,U)$. That is, paths on $S^2$ whose image in $S^2$ remains in $U$ while traversing the time interval $K$. So, if $K=I$ then an element $V(K,U)$ is just a path which remains within $U$ for all $t$, and if $K=\left[\frac{1}{4},\frac{3}{4}\right]$ then $V(K,U)$ is the set of paths which can do anything they like in the intervals $\left[0,\frac{1}{4}\right)$ and $\left(\frac{3}{4},1\right]$ but must then go towards and remain inside $U$ for the rest of the time.

You can make $K$ more complicated to really emphasise how wild elements in $V(K,U)$ can be and in particular, if $K$ is disconnected and $U$ is also disconnected, you can suggest how these cases affect what constraints we must put on elements in $V(K,U)$.