query: Let $(X,d)$ be a compact metric space. Then $(X,d)$ is both complete and bounded.
positive_passages.docid: 18040
positive_passages.text: Show that a compact metric space is bounded
negative_passages.docid: 88185
negative_passages.text: Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
negative_passages.docid: 98553
negative_passages.text: Infinite metric space has open set $U$ which is infinite and its complement is infinite
negative_passages.docid: 35573
negative_passages.text: A closed ball in a metric space is a closed set
negative_passages.docid: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 19111
negative_passages.text: If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
negative_passages.docid: 36566
negative_passages.text: Equivalence classes and metric spaces
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: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 118508
negative_passages.text: Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
negative_passages.docid: 155621
negative_passages.text: Finiteness of the dimension of a normed space and compactness
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 35652
negative_passages.text: Every compact metric space is complete
negative_passages.docid: 146485
negative_passages.text: Prove that the product space is a metric space.
negative_passages.docid: 106534
negative_passages.text: "Closed subsets of compact space are compact" with an alternative definition of covering
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed