query: Characterisation of compactness in $\mathbb{R}^n$
positive_passages.docid: 154240
positive_passages.text: Inverse of Heine–Cantor theorem
negative_passages.docid: 95234
negative_passages.text: Quasi-components in compact space
negative_passages.docid: 47251
negative_passages.text: Non-compactness of $\mathbb{R}$ with the cocountable topology
negative_passages.docid: 36631
negative_passages.text: Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
negative_passages.docid: 25795
negative_passages.text: Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$
negative_passages.docid: 80414
negative_passages.text: Countable Product of Sequentially Compact spaces is Sequentially Compact
negative_passages.docid: 47903
negative_passages.text: How to prove every closed interval in R is compact?
negative_passages.docid: 47810
negative_passages.text: $C(X)$ is separable when $X$ is compact?
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: 151425
negative_passages.text: Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?
negative_passages.docid: 65346
negative_passages.text: Characterisation of uniformly continuous function
negative_passages.docid: 66334
negative_passages.text: How is every subset of the set of reals with the finite complement topology compact?
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: 29474
negative_passages.text: Compactness of $\operatorname{Spec}(A)$
negative_passages.docid: 48162
negative_passages.text: Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?
negative_passages.docid: 117781
negative_passages.text: Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
negative_passages.docid: 48783
negative_passages.text: Prove that the sum of two compact sets in $\mathbb R^n$ is compact.