query: Prove that *every* subset of a metric space $M$ can be written as the intersection of open sets.
positive_passages.docid: 157753
positive_passages.text: Show that any set in a metric space can be written as the intersection of open sets
negative_passages.docid: 99645
negative_passages.text: Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
negative_passages.docid: 69528
negative_passages.text: Metrizability of quotient spaces of metric spaces
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: 61930
negative_passages.text: Intersection of Closed and Compact Set is Closed
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
negative_passages.docid: 125580
negative_passages.text: Proving that every countable metric space is disconnected?
negative_passages.docid: 151984
negative_passages.text: Proving the existence of a metric space
negative_passages.docid: 118508
negative_passages.text: Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
negative_passages.docid: 31686
negative_passages.text: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set
negative_passages.docid: 154978
negative_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: 153934
negative_passages.text: How to prove that a set R\Z is open
negative_passages.docid: 155999
negative_passages.text: How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$
negative_passages.docid: 59654
negative_passages.text: Show that in a discrete metric space, every subset is both open and closed.
negative_passages.docid: 78082
negative_passages.text: Dense subset of two Banach spaces also dense in the intersection