Take the 2-minute tour ×  Let X be a (finite dimensional)
manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$
where I = [0,1] is the closed interval. Is this space
paracompact? What if we fix a point x in X and consider the
pointed path space, is this space paracompact?  share|improve
this question  1 Answer 1  up vote 3 down vote accepted  What
topology do you want on $C^\infty(I,X)$? One natural topology,
the one of uniform convergence of all derivatives, is
metrizable, so paracompact.  share|improve this answer  Your
Answer