Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ?

We assume in addition that closed metric balls are compact.

A post without this extra assumption is here.

Solutions Collecting From Web of "Is a uniquely geodesic space contractible? I"

Let us assume in addition that closed metric balls in your metric space $X$ are compact (I do not know what to say in general). I will use the notation $g(p,q)$ for the unique geodesic in $X$ connecting $p$ to $q$ and regarded as a map from
an interval $[0,D]$ to $X$, where $D$ is the distance from $p$ to $q$.

Pick a point $x_0\in X$ and define a family of maps $f_s: X\to X$ by $f_s(x)$ defined to be the point on the geodesic $g(x_0, x)$ so that the distance from $x_0$ to $f_s(x)$ is $s$ times the distance from $x_0$ to $x$. Clearly, $f_1(x)=x$, $f_0(x)=x_0$ for all $x\in X$. What is not clear is if the map is continuous. I think, it would follow from continuity of the map sending $x$ to the geodesic $g(x_0, x)$ (where I use topology of uniform convergence on the set of geodesics). Suppose that I have a sequence $x_n\in X$ convergent to $x\in X$. I claim that such sequence always contains a subsequence so that the corresponding sequence of geodesics converges to $g$.

By the compactness assumption I made and the Arzela-Ascoli theorem, the sequence of geodesics $g_n$ (regarded as maps of course) converges (after passing to a subsequence) to a geodesic $g’$ connecting $x_0$ to $x$. (Usually the A-A theorem is applied to maps with a fixed domain. In our case, the sequence of domains is a sequence of intervals $[0, D_n]$ where $D_n$ is the distance form $x_0$ to $x_n$. To resolve this issue, take some $D\ge D_n, \forall n,$ and extend the maps $g_n$ to $[0,D]$ by constant maps on intervals $[D_n,D]$.)

Since $X$ is uniquely geodesic, $g’=g$. Hence, our subsequence in $g_n$ converges to $g$. Now, use the fact that in a compact metric space $G$ (the space of suitable geodesics in our case), a sequence $g_n$ converges to $g$ if and only if every convergent subsequence in $(g_n)$ converges to $g$. Thus, we proved that the sequence of geodesics $(g_n)$ converges to $g$ and, hence, the map from $X$ to the space of geodesics is continuous.