Metric on the unit circle Let $d$ be a metric on the unit circle $S^1$ which defines the usual topology. Let $D = \sup_{x, y \in S^1} d(x,y)$ be the diameter for $S^1$ under this metric.
Is it true that the map $d : S^1 \times S^1 \to [0,D]$ is an open map?
Thank you!

The map is not always open. Here is a simple counter-example.
The unit circle is homeomorphic to the following figure.

This figure is endowed with the euclidean metric of the plane, that we then transport to a metric on the unit circle using the homeomorphism.
Now consider the two points in red on the figure (or more exactly, their image on the unit circle by the homeomorphism). Let us call them $x$ and $y$. The distance will send any small neighborhood of $(x,y)$ to a set of the form $(\delta, \delta']$, with $\delta' < D$. 
Consequently, this map is not open.
