query: Show that a metric space $M$ is compact if and only if all infinite open cover of $M$ have a proper subcover.
positive_passages.docid: 154978
positive_passages.text: Some equivalent formulations of compactness of a metric space
negative_passages.docid: 33859
negative_passages.text: How to show that a disjoint sum of metric spaces is metrizable?
negative_passages.docid: 65976
negative_passages.text: in a topological space only finite subsets are compact sets
negative_passages.docid: 136108
negative_passages.text: Arbitrary intersection of closed, connected subsets of a compact space connected?
negative_passages.docid: 113227
negative_passages.text: Measure spaces are proper subsets
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: 16060
negative_passages.text: Showing this metric space is complete
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 26794
negative_passages.text: Metrizability of Hausdorff continuous image of compact metric space
negative_passages.docid: 92005
negative_passages.text: Existence of compact/complete metric on countable set
negative_passages.docid: 140570
negative_passages.text: Suppose $E\subseteq\mathbb{R}$ is closed and $K\subseteq\mathbb{R}$ is compact. Show that every open cover of $E\cap K$ has a finite subcover
negative_passages.docid: 148994
negative_passages.text: $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
negative_passages.docid: 17842
negative_passages.text: Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
negative_passages.docid: 7402
negative_passages.text: Showing a compact metric space has a countable dense subset
negative_passages.docid: 35573
negative_passages.text: A closed ball in a metric space is a closed set
negative_passages.docid: 9268
negative_passages.text: Proving that the Union of Two Compact Sets is Compact
negative_passages.docid: 147914
negative_passages.text: Closed image of locally compact space