This is a follow-up to my previous question, here: Is one condition of Banakh spaces redundant?. In the answer to the question, I was told that the condition is not redundant. However, I now want to know, if a metric space $M$ has the property that for every point $x \in M$ and every positive real number $r$, the $r$-sphere centered on $x$ is nonempty and of cardinality $2$, then is $M$ a Banakh space? That is, does that condition force every $r$-sphere to have diameter $2r$? Or is there some sort of exotic counterexample?
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Do you have a reference for the notion of "Banakh space"? – PatrickR Aug 18 '23 at 22:52
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A counterexample would be the graph of $y=e^x$, with the metric induced from the Euclidean metric on $\mathbb R^2$.
Edit: For that matter, you could take the graph of any (nonlinear) function so long as it has the property that every circle centered on the graph intersects the graph at exactly two points. For example, any nonlinear [double edit: continuous] strictly increasing function will satisfy this.
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Indeed any (nonlinear, continuous :P) increasing function works. – Noah Schweber Aug 18 '23 at 03:45