query: Proving that the unit square $\{(x,y) \in \mathbb{R^{2}} | 0<x,y<1\}$ is open in the metric topology on $\mathbb{R^{2}}$
positive_passages.docid: 27855
positive_passages.text: Show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric.
negative_passages.docid: 66239
negative_passages.text: Verifying that a certain collection of intervals of $\mathbb R$ forms a topology
negative_passages.docid: 6415
negative_passages.text: Distance function is continuous in topology induced by the metric
negative_passages.docid: 75829
negative_passages.text: Dimension of open subsets of $R^n$
negative_passages.docid: 131082
negative_passages.text: Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
negative_passages.docid: 12467
negative_passages.text: Why does this proof work: Closed unit ball in $C_0$ is not compact
negative_passages.docid: 107701
negative_passages.text: Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
negative_passages.docid: 44704
negative_passages.text: familly of Radially open sets in $\mathbb{R}^2$
negative_passages.docid: 6118
negative_passages.text: Homeomorphism of the real line-Topology
negative_passages.docid: 38033
negative_passages.text: A surjective linear map into a finite dimensional space is open
negative_passages.docid: 44194
negative_passages.text: How to show that the unit sphere is a topological manifold?
negative_passages.docid: 128730
negative_passages.text: Open subspaces of locally compact Hausdorff spaces are locally compact
negative_passages.docid: 84104
negative_passages.text: Definition of neighborhood and open set in topology
negative_passages.docid: 132641
negative_passages.text: Proving that Euclidean Space is a complete metric space
negative_passages.docid: 63701
negative_passages.text: An example of a regular function over an open set
negative_passages.docid: 124688
negative_passages.text: Question about a metric on $\mathbb{R}^2$
negative_passages.docid: 118049
negative_passages.text: Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$