query: Proof of the open interval $(-1,1)$ with respect to $*$ is a group
positive_passages.docid: 17510
positive_passages.text: Proving a set is an abelian group.
negative_passages.docid: 130081
negative_passages.text: Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
negative_passages.docid: 99464
negative_passages.text: Proof Concerning Indexed family of Sets and Intervals
negative_passages.docid: 141642
negative_passages.text: Closure of a certain subset in a compact topological group
negative_passages.docid: 21892
negative_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 39692
negative_passages.text: The length of an interval covered by an infinite family of open intervals
negative_passages.docid: 73248
negative_passages.text: Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
negative_passages.docid: 70264
negative_passages.text: Prove that the inverse image of an open set is open
negative_passages.docid: 89125
negative_passages.text: Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$
negative_passages.docid: 110217
negative_passages.text: $GL(n,K)$ is open in $M(n,K)$
negative_passages.docid: 79424
negative_passages.text: Don't understand proof that interior of a set is open
negative_passages.docid: 125962
negative_passages.text: The set $A'$ is closed in a $T_1$ space
negative_passages.docid: 66239
negative_passages.text: Verifying that a certain collection of intervals of $\mathbb R$ forms a topology
negative_passages.docid: 36921
negative_passages.text: An Open Interval and a Half-open Interval are not Homeomorphic
negative_passages.docid: 68724
negative_passages.text: $||T-I|| < 1$ implies that $T$ is invertible.
negative_passages.docid: 117781
negative_passages.text: Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
negative_passages.docid: 16654
negative_passages.text: Unions and Intersections of Open Sets are Open