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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 so, is it every $G_{\delta}$?

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

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, in this 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 so, is it every $G_{\delta}$?

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 so, is it every $G_{\delta}$?

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, in this post YCor pointed out that, in general, $\mathbb{H}(X)$ is dense in $(\mathbb{H}_{\operatorname{fin}}(X),d_{\mathbb{H}(X)})$ (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 so, is it every $G_{\delta}$?

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, in this 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 so, is it every $G_{\delta}$?