query: Show that $(\mathbb{R},d)$ is not complete , where $d(x,y) =|\arctan(x) - \arctan(y)|$.
positive_passages.docid: 26547
positive_passages.text: Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
negative_passages.docid: 42494
negative_passages.text: The boundary is a closed set
negative_passages.docid: 51227
negative_passages.text: Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
negative_passages.docid: 92588
negative_passages.text: Showing that $(C_0([0,\infty)), \|\cdot\|_\infty)$ is complete.
negative_passages.docid: 155579
negative_passages.text: Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
negative_passages.docid: 132641
negative_passages.text: Proving that Euclidean Space is a complete metric space
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: 80735
negative_passages.text: Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
negative_passages.docid: 46917
negative_passages.text: How to prove that $GL(n,\mathbb R)$ is not connected subset and open subset of$M_n (\mathbb{R})$
negative_passages.docid: 143105
negative_passages.text: Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
negative_passages.docid: 25874
negative_passages.text: Showing that a certain map is not flat by explicit counterexample
negative_passages.docid: 76148
negative_passages.text: Show that $\exists c \in \mathbb{R} : f(c) = 0$
negative_passages.docid: 149095
negative_passages.text: $f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
negative_passages.docid: 61962
negative_passages.text: Show $R$ is right-Artinian but not left-Artinian
negative_passages.docid: 35569
negative_passages.text: Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
negative_passages.docid: 77464
negative_passages.text: Prove that ${x dy - y d x\over x^2 + y^2}$ is not exact
negative_passages.docid: 116768
negative_passages.text: Problem 5, sec. 13 in Munkres' Topology, 2nd ed.: How to prove the assertion if $\mathscr{A}$ is a subbasis?