query: In the definition of 'Compact set', why do we require the existence of finite subcover for any open cover? Why not countable subcover?
positive_passages.docid: 73260
positive_passages.text: Why is compactness so important?
negative_passages.docid: 80180
negative_passages.text: Discrete and compact subset must be finite
negative_passages.docid: 59275
negative_passages.text: If $X$ is an infinite set, then $X$ with the discrete topology is not 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: 102543
negative_passages.text: Real definition of "countable set"
negative_passages.docid: 140168
negative_passages.text: What are the basis for co-countable and co-finite topologies
negative_passages.docid: 73130
negative_passages.text: A simple question about open set
negative_passages.docid: 16826
negative_passages.text: prove that every infinite subset of compact set H has a limit point (Explanation)
negative_passages.docid: 94825
negative_passages.text: Open set as a finite union of closed sets
negative_passages.docid: 128437
negative_passages.text: How to prove that a finite set is countable
negative_passages.docid: 47810
negative_passages.text: $C(X)$ is separable when $X$ is compact?
negative_passages.docid: 45249
negative_passages.text: Pseudocompactness does not imply compactness
negative_passages.docid: 148507
negative_passages.text: Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
negative_passages.docid: 88099
negative_passages.text: Not $\sigma$-compact set without axiom of choice
negative_passages.docid: 111008
negative_passages.text: Show that $X $ is a infinite set ,it is connected in the finite complement topology?
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: 74860
negative_passages.text: Why is the empty set a subset of every set?