I have a question reading the proof of Lemma 16.1 in Milnor's "Morse Theory", pp.88-89. https://www.maths.ed.ac.uk/~v1ranick/papers/milnmors.pdf
This lemma is asserting that if $M$ is a complete Riemannian manifold and $\Omega=\Omega(M;p,q)$ is the set of all piecewise smooth curves in $M$ from $p\in M$ to $q\in M$, then a certain subset of $\Omega$ can be given a structure of a (finite-dimensional) smooth manifold.
(All the notations are given in p.88 of the book. I think it will be too long if I write the whole page to explain the notations.)
At the last paragraph of the proof, Milnor defines a map $\text{Int} \Omega(t_0,t_1,\dots,t_k)^c \to M^{k-1}=M\times \cdots \times M$ by $\omega\mapsto (\omega(t_1),\dots,\omega(t_{k-1}))$. (Let us call this map as $F$.) The argument in the proof shows that $F$ is injective. Also, by the definition of the distance function on $\Omega$, $F$ is certainly continuous.
What I can't show is:
(1) The image of $F$ is an open subset of $M^{k-1}$.
(2) $F$ is a homeomorphism onto its image.
Milnor says that these are evident, but I can't see why.