query: "Closed subsets of compact space are compact" with an alternative definition of covering
positive_passages.docid: 146146
positive_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 51156
negative_passages.text: Function is Continuous When Restricted to Closed Sets Which Cover the Space
negative_passages.docid: 13848
negative_passages.text: Showing that if a subset of a complete metric space is closed, it is also complete
negative_passages.docid: 28391
negative_passages.text: Prove the compactness of a set B
negative_passages.docid: 158814
negative_passages.text: Two definitions of locally compact space
negative_passages.docid: 82900
negative_passages.text: Show that Minkowski sum of a closed set and a compact set in $\mathbb{R}^n$ is closed.
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 111360
negative_passages.text: Topological spaces in which every proper closed subset is compact
negative_passages.docid: 73353
negative_passages.text: Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
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: 60424
negative_passages.text: Uncountable family of uncountable compact subsets of $\mathbb{R}$
negative_passages.docid: 481
negative_passages.text: Can compact sets completey determine a topology?
negative_passages.docid: 145415
negative_passages.text: Examples of almost compact spaces.
negative_passages.docid: 55648
negative_passages.text: Intersection of finite number of compact sets is compact?
negative_passages.docid: 73154
negative_passages.text: Compact subset in colimit of spaces
negative_passages.docid: 7402
negative_passages.text: Showing a compact metric space has a countable dense subset
negative_passages.docid: 106957
negative_passages.text: Compact sets are closed?