In one of my books about metric spaces, it is stated that "a not null set, equiped with a metric, is a metric space". On the other hand, the null set as a subset of $R$ is a compact subspace as it is complete and totally bounded.... Why is there such a contradiction?Thanks.
2026-03-26 17:30:47.1774546247
The null space in metric spaces
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The author decided to require metric spaces to be non-empty. This is not a definition I would adopt (see Why are metric spaces non-empty? for example) but it does not contradict anything.
If we follow this definition, then $\varnothing$ is still a compact subset of $\mathbb{R}$, but it is not recognized as a metric space of its own. Not a contradiction.