In the some of article from [Michael Farber](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&sqi=2&ved=0ahUKEwiq772cspXTAhXEKJoKHaDxA14QFggeMAA&url=https%3A%2F%2Farxiv.org%2Fabs%2Fmath%2F0111197&usg=AFQjCNGsyuehFyRcIq8928HY3GKIqPikJQ&sig2=kDgOiM_B--V35fODJkKMYQ&bvm=bv.152174688,d.d24) that is about topological complexity, I'm stuck in continuity of $S\_1$ as follows:  

Define $U\_1 ⊂ S^1 \times S^1$
as $U\_1 = \{(A, B); A \neq -B\}$ .A continuous motion planning over $U\_1$ is given by the map $S\_1: U\_1 → PS^1$ which moves $A$ towards $B$ with constant velocity along the unique shortest arc connecting $A$ to $B$.  

Equip the path space $PS^1$ with compact-open topology,i can't understand why $S\_1$ is continuous!  

For more information you can refer to My [previous question](https://mathoverflow.net/questions/246784/applications-of-topological-complexity-of-configuration-space)  

Can somebody halp me?


Thanks. 



Topological complexity of circle
If $(X,d)$ is a metric space, then the compact open topology on $PX$ is metrizable, and is induced by the metric $$\rho (\gamma, \omega) =\operatorname {sup}\{ d (\gamma (t), \omega (t) )\mid t\in I\}. $$


Knowing this, most people would take continuity of the map you describe as self-evident. If you want a proof, however, it shouldn't be too hard to write one using the $\epsilon$-$\delta $ definition of continuity.


