query: Subspace of a separable metric space is also separable
positive_passages.docid: 19856
positive_passages.text: Prove that a subset of a separable set is itself separable
negative_passages.docid: 96717
negative_passages.text: Hahn-Banach Theorem for separable spaces without Zorn's Lemma
negative_passages.docid: 43504
negative_passages.text: Proving the existence of a complement subspace $U' \subset V$.
negative_passages.docid: 36566
negative_passages.text: Equivalence classes and metric spaces
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 8334
negative_passages.text: A reflexive Banach space is separable iff its dual is separable
negative_passages.docid: 112303
negative_passages.text: Distances between closed sets on metric spaces
negative_passages.docid: 84791
negative_passages.text: Is every connected metric space with at least two points uncountable?
negative_passages.docid: 17842
negative_passages.text: Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
negative_passages.docid: 108619
negative_passages.text: Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
negative_passages.docid: 146967
negative_passages.text: Understanding equivalent metric spaces
negative_passages.docid: 125073
negative_passages.text: Finite-dimensional subspaces of normed vector spaces are direct summands
negative_passages.docid: 157565
negative_passages.text: Is every dense subspace of a separable space separable?
negative_passages.docid: 150987
negative_passages.text: Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent
negative_passages.docid: 125580
negative_passages.text: Proving that every countable metric space is disconnected?
negative_passages.docid: 154978
negative_passages.text: Some equivalent formulations of compactness of a metric space
negative_passages.docid: 79779
negative_passages.text: How big can a separable Hausdorff space be?