query: Let $(X,d)$ be a metric space and $ A\subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$
positive_passages.docid: 45581
positive_passages.text: Is the distance function in a metric space (uniformly) continuous?
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 22131
negative_passages.text: Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
negative_passages.docid: 128894
negative_passages.text: About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
negative_passages.docid: 51914
negative_passages.text: Show that $S$ is compact.
negative_passages.docid: 22854
negative_passages.text: Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 9406
negative_passages.text: Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
negative_passages.docid: 35652
negative_passages.text: Every compact metric space is complete
negative_passages.docid: 143484
negative_passages.text: A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
negative_passages.docid: 104068
negative_passages.text: Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$
negative_passages.docid: 103451
negative_passages.text: Show that $Y$ is locally compact.
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 111119
negative_passages.text: Distance between any two points in a compact metric space
negative_passages.docid: 51227
negative_passages.text: Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
negative_passages.docid: 134914
negative_passages.text: Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
negative_passages.docid: 42266
negative_passages.text: Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$