query: In $(X,d)$ metric space, an intersection of finite families of open sets is open.
positive_passages.docid: 16654
positive_passages.text: Unions and Intersections of Open Sets are Open
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: 123753
negative_passages.text: In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
negative_passages.docid: 151984
negative_passages.text: Proving the existence of a metric space
negative_passages.docid: 70048
negative_passages.text: In which topologies do open sets maintain open under countable or arbitrary intersection?
negative_passages.docid: 61930
negative_passages.text: Intersection of Closed and Compact Set is Closed
negative_passages.docid: 124179
negative_passages.text: Proof that every metric space is normal
negative_passages.docid: 114061
negative_passages.text: Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 98553
negative_passages.text: Infinite metric space has open set $U$ which is infinite and its complement is infinite
negative_passages.docid: 7607
negative_passages.text: How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
negative_passages.docid: 55648
negative_passages.text: Intersection of finite number of compact sets is compact?
negative_passages.docid: 49721
negative_passages.text: Prove the empty set is closed for a metric space (X,d).
negative_passages.docid: 44623
negative_passages.text: Closed set as a countable intersection of open sets
negative_passages.docid: 21570
negative_passages.text: Intersection of a family of non-empty closed intervals
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: 88185
negative_passages.text: Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed