query: A strange criterion for compactness
positive_passages.docid: 19111
positive_passages.text: If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
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: 41408
negative_passages.text: Which of the following sets are compact:
negative_passages.docid: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 19566
negative_passages.text: Continuous images of compact sets are compact
negative_passages.docid: 6210
negative_passages.text: Countable compact spaces as ordinals
negative_passages.docid: 148187
negative_passages.text: Question regarding one-point compactification and homeomorphisms
negative_passages.docid: 37654
negative_passages.text: Topology: Example of a compact set but its closure not compact
negative_passages.docid: 80202
negative_passages.text: Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$?
negative_passages.docid: 40682
negative_passages.text: What's going on with "compact implies sequentially compact"?
negative_passages.docid: 63529
negative_passages.text: How to show that this set is compact in $\ell^2$
negative_passages.docid: 83311
negative_passages.text: Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
negative_passages.docid: 38840
negative_passages.text: Tightness and Inner Regularity
negative_passages.docid: 50611
negative_passages.text: Question about Stone-Čech compactification
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
negative_passages.docid: 144400
negative_passages.text: Are compact subsets of metric spaces closed and bounded?
negative_passages.docid: 7402
negative_passages.text: Showing a compact metric space has a countable dense subset