query: Prove that if $A,B$ are closed and disjoint in a metric space, then they are contained in disjoint neighborhoods
positive_passages.docid: 27998
positive_passages.text: Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
negative_passages.docid: 52014
negative_passages.text: Showing $(C[0,1], d_1)$ is not a complete metric space
negative_passages.docid: 36237
negative_passages.text: Closed set mapped to itself in a compact Hausdorff space
negative_passages.docid: 117508
negative_passages.text: Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$
negative_passages.docid: 144383
negative_passages.text: The closure of a connected set in a topological space is connected
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: 129243
negative_passages.text: Is $C_{c}^{\infty}[a,b]$ a complete metric space?
negative_passages.docid: 114599
negative_passages.text: $A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected
negative_passages.docid: 109682
negative_passages.text: Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 112303
negative_passages.text: Distances between closed sets on metric spaces
negative_passages.docid: 90628
negative_passages.text: The metric spaces are Tychonoff spaces
negative_passages.docid: 106756
negative_passages.text: Union and Intersection of Sets $A_{i}$ and $B_{j}$
negative_passages.docid: 110744
negative_passages.text: How to prove that a compact set in a Hausdorff topological space is closed?
negative_passages.docid: 119105
negative_passages.text: If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
negative_passages.docid: 80276
negative_passages.text: Metric Space & Cauchy Sequence
negative_passages.docid: 116069
negative_passages.text: Is every metric space a subspace of some connected metric space?