Intereting Posts

Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.
Limit of sequence of growing matrices
Meaning of algebraic decay
Logical implication help
Abel limit theorem
Prove that H is a subgroup of G.
When is a divisible group a power of the multiplicative group of an algebraically closed field?
Right invertible and left zero divisor in matrix rings over a commutative ring
Find inverse Laplace Transform using s-shifting and t-shifting. (5.3-57)
Why is in the category of pointed sets not every epimorphism a cokernel?
About fractional iterations and improper integrals
Why can the transformation derived from a list of points and a list of their transformed counterparts not be affine or linear?
Determine if the following series are convergent or divergent?
What is the general solution to the first order differential equation $dy/dx =(x^3 – 2y)/x$?
Indiscrete rational extension for $\mathbb R$ (examp 66 in “Counterexamples in topology”)

Is a uniquely geodesic space contractible ?

We assume in addition that closed metric balls are compact.

A post without this extra assumption is here.

- Abelianization of free product is the direct sum of abelianizations
- Applications of algebraic topology
- Homology of a co-h-space manifold
- Homology of $S^1 \times (S^1 \vee S^1)$
- $H$ a discrete subgroup of topological group $G$ $\implies$ there exists an open $U\supseteq\{1\}$ s.t. the $hU$ are pairwise disjoint
- Conceptualizing Inclusion Map from Figure Eight to Torus

- Cofinite\discrete subspace of a T1 space?
- Torus as double cover of the Klein bottle
- Intuitive Aproach to Dolbeault Cohomology
- Fundamental group and path-connected
- Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $
- Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).
- Proving that the union of the limit points of sets is equal to the limit points of the union of the sets
- When is a notion of convergence induced by a topology?
- Homology of connected sum of real projective spaces
- Congruent division of a shape in euclidean plane

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.

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