query: Every open set as union of closed sets
positive_passages.docid: 152976
positive_passages.text: Real Analysis: open and closed sets
negative_passages.docid: 30231
negative_passages.text: The boundary of union is the union of boundaries when the sets have disjoint closures
negative_passages.docid: 63276
negative_passages.text: Is an arbitrary union of a chain of countable sets countable?
negative_passages.docid: 114608
negative_passages.text: Proof concerning regular space: there exist a closed set contained in any open set containing $x$
negative_passages.docid: 99236
negative_passages.text: Interior of closure of an open set
negative_passages.docid: 91991
negative_passages.text: How do I prove that two closed sets are separable?
negative_passages.docid: 33310
negative_passages.text: Proving that the set of limit points of a set is closed
negative_passages.docid: 70264
negative_passages.text: Prove that the inverse image of an open set is open
negative_passages.docid: 110957
negative_passages.text: No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
negative_passages.docid: 6491
negative_passages.text: Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls.
negative_passages.docid: 152193
negative_passages.text: The intersection of a closure with an open set
negative_passages.docid: 2198
negative_passages.text: Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
negative_passages.docid: 72468
negative_passages.text: Open and closed balls in $C[a,b]$
negative_passages.docid: 24372
negative_passages.text: Infinite Set is Disjoint Union of Two Infinite Sets
negative_passages.docid: 143457
negative_passages.text: Is product of two closed sets closed?