query: Showing a metric space to be incomplete
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: 35804
negative_passages.text: Bi-Lipschitz map preserves completeness
negative_passages.docid: 128897
negative_passages.text: Bounded sequence in a metric space
negative_passages.docid: 17067
negative_passages.text: To prove that $(\mathbb{P(N)},d)$ is a metric space
negative_passages.docid: 66550
negative_passages.text: Cardinality of separable metric spaces
negative_passages.docid: 11455
negative_passages.text: Connected metric spaces with at least 2 points are uncountable.
negative_passages.docid: 84004
negative_passages.text: Prove there is no contraction mapping from compact metric space onto itself
negative_passages.docid: 72950
negative_passages.text: Book recommendation for Metric Spaces
negative_passages.docid: 94438
negative_passages.text: Show that the countable product of metric spaces is metrizable
negative_passages.docid: 126759
negative_passages.text: Unbounded continuous function on non-compact metric space
negative_passages.docid: 107321
negative_passages.text: How to show convexity of a ball in metric space?
negative_passages.docid: 47820
negative_passages.text: How can I show a supremum of a set is also its limit point?
negative_passages.docid: 129243
negative_passages.text: Is $C_{c}^{\infty}[a,b]$ a complete metric space?
negative_passages.docid: 146485
negative_passages.text: Prove that the product space is a metric space.
negative_passages.docid: 65840
negative_passages.text: Norm induced by metric
negative_passages.docid: 149416
negative_passages.text: Show that $ \{\lnot,\leftrightarrow\} $ is not functional complete
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space