query: Is continuum $ 2^{\mathbb{R}}$?
positive_passages.docid: 52682
positive_passages.text: Is the power set of the natural numbers countable?
negative_passages.docid: 66272
negative_passages.text: Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
negative_passages.docid: 154594
negative_passages.text: Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$?
negative_passages.docid: 69869
negative_passages.text: if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
negative_passages.docid: 121708
negative_passages.text: $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 22890
negative_passages.text: Is it coherent to extend $\mathbb{R}$ with a reciprocal of $0$?
negative_passages.docid: 33330
negative_passages.text: Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
negative_passages.docid: 35305
negative_passages.text: Why is $\mathbb{R}^2$ not a subset and /or a subspace of $\mathbb{R}^3$?
negative_passages.docid: 63598
negative_passages.text: Is every separately continuous function on $R^2$ continuous?
negative_passages.docid: 40071
negative_passages.text: What is the cardinality of the set of all topologies on $\mathbb{R}$?
negative_passages.docid: 64920
negative_passages.text: Boundary Question in $\mathbb{R}^{2}$ (Manifolds)
negative_passages.docid: 117419
negative_passages.text: Why is $e^{x}$ not uniformly continuous on $\mathbb{R}$?
negative_passages.docid: 61375
negative_passages.text: Meaning of $\mathbb{R}[x]$
negative_passages.docid: 121198
negative_passages.text: $W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$?
negative_passages.docid: 35626
negative_passages.text: Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$?