Let $K$ be a compact metric space. Now remove one point from it. Is there a metric on $K$ with one point removed that generates the same topology as the initial metric but is such that the space $K$ with one point removed is still complete?
2026-04-04 15:32:43.1775316763
Metric on compact set with one point less
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If $(K,d)$ is any complete metric space and $x \in K$ then $D(u,v)=d(u,v)+|\frac 1 {d(u,x)} -\frac 1 {d(v,x)}|$ makes $K\setminus \{x\}$ a complete metric space and $d$ is equivalent to $D$ on $K\setminus \{x\}$.