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This is a follow-up to my previous questions, here: Is one condition of Banakh spaces redundant? and A follow-up to a question on Banakh spaces. Refer to the first question for the definition of a Banakh space. I define a strong Banakh space to be a non-empty metric space $M$ such that for any point $x$ in $M$ and any positive real number $r$, the sphere $\{y:d(x,y)=r\}$ has cardinality $2$ and diameter $2r$. This is indeed a strengthening of the usual definition of Banakh space. My question is, is every Banakh space isometric to the real line, or at least a subset of the real line?

user107952
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    Worth noting that a proper subset of the real line will never satisfy your condition - pick any point in the subset, for all its spheres to have cardinality two it must be all of $\mathbb R$. – M W Oct 17 '23 at 13:56
  • Stefan Banach wasn't named Banakh – Didier Oct 17 '23 at 18:49
  • @Didier pretty sure Banakh spaces have nothing to do with Stefan Banach. – M W Oct 17 '23 at 21:21
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    @MW You must be right. Sorry, never ever heard about Banakh spaces actually. What a coincidental name – Didier Oct 17 '23 at 21:24
  • Seems this mathoverflow discussion suggests answer is yes if you add completeness, no otherwise. https://mathoverflow.net/questions/442772/a-metric-characterization-of-the-real-line/442872#442872. (I think this discussion was the origin of the concept of a Banakh space in the first place) – M W Oct 17 '23 at 21:29

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