query: Show a non-empty open and closed set in R must be equal to R
positive_passages.docid: 11405
positive_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 148507
negative_passages.text: Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
negative_passages.docid: 158569
negative_passages.text: Proving a particular subset of $R^n$ is closed
negative_passages.docid: 42145
negative_passages.text: Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
negative_passages.docid: 47680
negative_passages.text: Prove that the union of the interior of a set and the boundary of the set is the closure of the set
negative_passages.docid: 85225
negative_passages.text: The set of integers is not open or is open
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 138202
negative_passages.text: Every open set in $\mathbb{R}$ is a countable union of closed sets
negative_passages.docid: 6291
negative_passages.text: Proving something is a linearly independent subset
negative_passages.docid: 110957
negative_passages.text: No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
negative_passages.docid: 39293
negative_passages.text: Given a compact set $K$ and closed set $E,$ prove: $E\cap K\neq \emptyset $
negative_passages.docid: 62208
negative_passages.text: If $A$ is a non-empty set, then $A \notin A$
negative_passages.docid: 126897
negative_passages.text: Showing the set $A+B$ is closed.
negative_passages.docid: 138283
negative_passages.text: $|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval
negative_passages.docid: 48797
negative_passages.text: Proving Two Sets are Equal - Infinite Sets - Example
negative_passages.docid: 41611
negative_passages.text: Open and closed set in n by n matrices space.
negative_passages.docid: 82628
negative_passages.text: Cardinality of a set of closed intervals