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Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(X)$ denote the subset of $\mathbb{H}(X)$ consisting of non-empty finite sets.

If $X$ is finite then clearly $\mathbb{H}(X)=\mathbb{H}_{\operatorname{fin}}(X)$. Moreover, a previous post, YCor pointed out that, in general, $\mathbb{H}_{\operatorname{fin}}(X)$ is dense in $(\mathbb{H}(X),d_{\mathbb{H}(X)})$ (see this MSE post for a short and simple proof).

The discussion below the first linked post inspired the following "coupled questions":

  • Is $\mathbb{H}_{\operatorname{fin}}(X)$ ever residual in $X$, when $X$ is infinite?

  • If not: Is there a "standard" (weaker) topology on $\mathbb{H}(X)$ for which $\mathbb{H}_{\operatorname{fin}}(X)$ is residual

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    $\begingroup$ The set of nonempty finite subsets of cardinal $\le n$ is closed. Moreover, if $X$ has no isolated point, then it has empty interior. Hence if $X$ has no isolated point, the set of nonempty finite subsets is an $F_\sigma$ of empty interior (i.e., the complement of a $G_\delta$-dense subset, i.e., of a residual subset). So it's not residual (for $X$ nonempty with no isolated point). $\endgroup$
    – YCor
    Commented Jun 13, 2022 at 22:44
  • $\begingroup$ @YCor I also modified the post to further ask, if there is a "typical/standard" topology on $\mathbb{H}(X)$ for which the collection of finite sets is residual? $\endgroup$
    – ABIM
    Commented Jun 14, 2022 at 8:01
  • $\begingroup$ The hyperspace of finite subset is dense in the hyperspace of compact subsets but not in the hyperspace of all closed sets (because the latter is not separable for non-compact $X$). $\endgroup$
    – Taras Banakh
    Commented Jun 18, 2022 at 3:37

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