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To elaborate, if I were asked to prove that a set $S$ is closed, and I know that there exists a particular metric space $(X,d)$ such that $S\subset X$, is it then sufficient to prove that $S$ is closed in $X$, using the particular metric $d$?

joshuaheckroodt
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  • Firstly, $S \subseteq X$ is a fact about sets, not metrics. If $S \subseteq X$ it is true regardless of the metric.

    As for your other question- closure depends on the metric. A set may be closed under one metric but not under another. If the metric is not specified then you're probably supposed to assume some "standard" metric.

     – Vercingetorix Feb 12 '21 at 10:02

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If no metric is given, then the question makes no sense, even if you are aware of the existence of such a metric space. If someone asks you whether or not $(0,1)$ is closed, what's your answer? Yes, $(0,1)\subset\Bbb R$, but it is closed with respect to the discrete metric, whereas it is not closed with respect to the usual one.

José Carlos Santos
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