query: Show that $\mathbb{R} \backslash \mathbb{Z}$ is open.
positive_passages.docid: 153934
positive_passages.text: How to prove that a set R\Z is open
negative_passages.docid: 135611
negative_passages.text: Extensions of $\mathbb{Z}$ by $\mathbb{Z}$
negative_passages.docid: 83001
negative_passages.text: Easier way to show $(\mathbb{Z}/(n))[x]$ and $\mathbb{Z}[x] / (n)$ are isomorphic
negative_passages.docid: 79424
negative_passages.text: Don't understand proof that interior of a set is open
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 152976
negative_passages.text: Real Analysis: open and closed sets
negative_passages.docid: 13162
negative_passages.text: Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
negative_passages.docid: 62755
negative_passages.text: The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
negative_passages.docid: 131082
negative_passages.text: Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
negative_passages.docid: 121465
negative_passages.text: $R[x]$ has a subring isomorphic to $R$.
negative_passages.docid: 21599
negative_passages.text: $(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
negative_passages.docid: 94825
negative_passages.text: Open set as a finite union of closed sets
negative_passages.docid: 1020
negative_passages.text: Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain
negative_passages.docid: 50344
negative_passages.text: What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$?
negative_passages.docid: 92517
negative_passages.text: Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$
negative_passages.docid: 136106
negative_passages.text: Is every open subset of $ \mathbb{R} $ uncountable?
negative_passages.docid: 146470
negative_passages.text: Show $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain