Let $(X,d)$ be a metric space and $x_0$ be an element in $X$. Define $f:X \to \mathbb R$ by $f(x)=d(x_0,x)$. Does this map send closed sets to closed sets? Further, let $C$ be a closed subset of $X$. Is $d(x_0,C) \in f(C)$?
2026-03-25 06:18:50.1774419530
Does the distance map send closed sets to closed sets?
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No. Let X = $\{-1\} \cup \{1/n : n \text{ a positive integer} \}$, let $C = \{ 1/n : n \text{ a positive integer } \}$, $x_0 = -1$.
$X$ with the inherited metric is a discrete space. $C$ is a closed subset of $X$, but $f(C) = \{ 1 + 1/n : n \text{ a positive integer } \}$ isn't a closed subset of $\mathbb R$. Further, $d(-1, C) = 1 \notin f(C)$.