query: It has been shown that every open set in ℝ is the union of disjoint open intervals. Is this true for $ℝ^n$ in general?
positive_passages.docid: 148507
positive_passages.text: Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
negative_passages.docid: 108394
negative_passages.text: Proof of that every interval is connected
negative_passages.docid: 156520
negative_passages.text: Intervals are connected and the only connected sets in $\mathbb{R}$
negative_passages.docid: 63701
negative_passages.text: An example of a regular function over an open set
negative_passages.docid: 14215
negative_passages.text: Show that the union $\bigcup A_n$ has empty interior in $X$
negative_passages.docid: 27855
negative_passages.text: Show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric.
negative_passages.docid: 106944
negative_passages.text: Is the set defined by y=sin(1/x) open or closed?
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 138283
negative_passages.text: $|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval
negative_passages.docid: 109046
negative_passages.text: Is the complement of an open ball in a Banach space connected?
negative_passages.docid: 2198
negative_passages.text: Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous
negative_passages.docid: 123149
negative_passages.text: Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
negative_passages.docid: 129081
negative_passages.text: Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
negative_passages.docid: 146664
negative_passages.text: The union of two linearly independent subsets
negative_passages.docid: 36315
negative_passages.text: Is $[0,1]$ a countable disjoint union of closed sets?
negative_passages.docid: 24372
negative_passages.text: Infinite Set is Disjoint Union of Two Infinite Sets
negative_passages.docid: 72468
negative_passages.text: Open and closed balls in $C[a,b]$