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Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$).

Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open balls $B(x,r)=\{y \in \Sigma, d(x,y)<r\}$ are homeomorphic to an open disc on the plane.

Then, is the following true: for every $r<1$, the closed balls $\overline{B}(x,r)=\{y \in \Sigma,d(x,y) \leq r\}$ are homeomorphic to a closed disk on the plane ? (since the metric is intrisic, $\overline{B}(x,r)$ is the closure of $B(x,r)$).

I think it is true, but I can not find any proof. We can see this as a problem about the topology of closed sets in the plane, but I don't know much about it.

You can write $\overline{B}(x,r)$ as the non-increasing intersection of the open balls $B(x,s)$ for $s>r$, which are homeomorphic to an open disc; maybe this could help...

PS : you can find easy examples showing that neither "$B(x,r)$ is homeomorphic to an open disc" nor "$\overline{B}(x,r)$ is homeomorphic to a closed disc" is a consequence of the other.

An idea might be to find a characterization of a set homeomorphic to a closed disc. For example with a property of its interior, or its boundary, or its homotopy groups, or something else... I would be very interested by any idea!

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  • $\begingroup$ What about round sphere of radius $1/2\pi$? $\endgroup$ – Alex Degtyarev Apr 7 '16 at 16:55
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    $\begingroup$ It verifies both statement. But I'd like a proof that "statement 1" implies "statement 2" for every surface and every metric... $\endgroup$ – Clem. Apr 7 '16 at 18:21
  • $\begingroup$ This is true; are you still interested in a proof? $\endgroup$ – Moishe Kohan Dec 1 '16 at 5:34
  • $\begingroup$ Yes I am interested! $\endgroup$ – Clem. Dec 4 '16 at 9:35
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This is really a comment but I am not entitled. There is a purely topological characterization of $n$-cubes due to J. de Groot in the Felix Hausdorff Gedenkband (1972): Topological characterization of metrizable cubes. This specialises to one for the closed disc as requested in your question. The condition is too intricate for me to quote it here but can be easily accessed at the review of this article (MR0348723).

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