7
$\begingroup$

This question deals with a special case of this question, which has not yet been satisfactorily solved. If you have any ideas about that general case, feel free to answer there and I'll be happy to close/delete this question.

Let $(X,d)$ be a metric space such that every open set is the disjoint union of open balls. Also assume that $(X,d)$ is a length space, i.e. for all $x,y\in X$ and $L>d(x,y)$ there exists a weakly contracting curve $\gamma\colon [0,L]\to X$ from $x$ to $y$ (that is, we have $\gamma(0)=x$, $\gamma(L)=y$ and $d(\gamma(t_1),\gamma(t_2))\le |t_1-t_2|$ for $t_1,t_2\in[0,L]$).

Conjecture: $(X,d)$ is homeomorphic to a connected subspace of $S^1$ (that is, one of $\{0\}$, $(0,1)$, $[0,1)$, $[0,1]$, $S^1$.

Note that the more general question was about connected spaces and that length spaces are connected. I have a strong feeling that the case of length spaces is a lot simpler.

1 Answers 1