query: Is the Axiom of Choice necessary in the proof that any subset of a metric space that contains its limit points is closed?
positive_passages.docid: 154578
positive_passages.text: Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 27998
negative_passages.text: Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
negative_passages.docid: 121135
negative_passages.text: Why don't we use closed covers to define compactness of metric space?
negative_passages.docid: 106534
negative_passages.text: "Closed subsets of compact space are compact" with an alternative definition of covering
negative_passages.docid: 64260
negative_passages.text: Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
negative_passages.docid: 144899
negative_passages.text: Do We Need the Axiom of Choice for Finite Sets?
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 78117
negative_passages.text: Closed property of nonempty finite set
negative_passages.docid: 28629
negative_passages.text: Is the axiom of choice necessary to prove that closed points in the Zariski topology are maximal ideals?
negative_passages.docid: 33310
negative_passages.text: Proving that the set of limit points of a set is closed
negative_passages.docid: 24050
negative_passages.text: Characterization of the Closure of a Set
negative_passages.docid: 10180
negative_passages.text: How to prove that this set is closed?
negative_passages.docid: 108619
negative_passages.text: Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
negative_passages.docid: 16826
negative_passages.text: prove that every infinite subset of compact set H has a limit point (Explanation)
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space
negative_passages.docid: 154978
negative_passages.text: Some equivalent formulations of compactness of a metric space