query: is intersection of a countable collection of dense, open subsets of a complete metric space also dense in X?
positive_passages.docid: 27400
positive_passages.text: About Baire's Category Theorem(BCT)
negative_passages.docid: 137881
negative_passages.text: A not complete metric space?
negative_passages.docid: 84791
negative_passages.text: Is every connected metric space with at least two points uncountable?
negative_passages.docid: 49721
negative_passages.text: Prove the empty set is closed for a metric space (X,d).
negative_passages.docid: 95831
negative_passages.text: Compactness and sequential compactness in metric spaces
negative_passages.docid: 68230
negative_passages.text: The cardinality of a countable union of sets with less than continuum cardinality
negative_passages.docid: 90590
negative_passages.text: Prove that the following set is dense
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: 106455
negative_passages.text: intersection of infinite collection of finite sets?
negative_passages.docid: 18039
negative_passages.text: Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the induced subspace topology, then $X$ is also connected
negative_passages.docid: 55648
negative_passages.text: Intersection of finite number of compact sets is compact?
negative_passages.docid: 107701
negative_passages.text: Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
negative_passages.docid: 127678
negative_passages.text: A partition of the unit interval into uncountably many dense uncountable subsets
negative_passages.docid: 33859
negative_passages.text: How to show that a disjoint sum of metric spaces is metrizable?
negative_passages.docid: 2403
negative_passages.text: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 106434
negative_passages.text: Prove that for a dense set $A$ and an open set $U$, $U\subset \overline{A\cap U}$