query: If $K$ is closed and bounded, then it is compact.
positive_passages.docid: 148994
positive_passages.text: $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 95831
negative_passages.text: Compactness and sequential compactness in 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: 63529
negative_passages.text: How to show that this set is compact in $\ell^2$
negative_passages.docid: 41408
negative_passages.text: Which of the following sets are compact:
negative_passages.docid: 9268
negative_passages.text: Proving that the Union of Two Compact Sets is Compact
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
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: 61930
negative_passages.text: Intersection of Closed and Compact Set is Closed
negative_passages.docid: 94485
negative_passages.text: Limit point compactness implies compactness.
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 55648
negative_passages.text: Intersection of finite number of compact sets is compact?
negative_passages.docid: 48162
negative_passages.text: Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?
negative_passages.docid: 36631
negative_passages.text: Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
negative_passages.docid: 59194
negative_passages.text: Spaces in which "$A \cap K$ is closed for all compact $K$" implies "$A$ is closed."
negative_passages.docid: 74311
negative_passages.text: Proving that the Intersection of any Collection of Compact Sets is Compact