query: Show that every compact subspace of a metric space is bounded in that metric and is closed
positive_passages.docid: 144400
positive_passages.text: Are compact subsets of metric spaces closed and bounded?
negative_passages.docid: 82333
negative_passages.text: let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
negative_passages.docid: 112303
negative_passages.text: Distances between closed sets on metric spaces
negative_passages.docid: 144353
negative_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 43431
negative_passages.text: Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.
negative_passages.docid: 52014
negative_passages.text: Showing $(C[0,1], d_1)$ is not a complete metric space
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 92005
negative_passages.text: Existence of compact/complete metric on countable set
negative_passages.docid: 124179
negative_passages.text: Proof that every metric space is normal
negative_passages.docid: 99645
negative_passages.text: Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 49721
negative_passages.text: Prove the empty set is closed for a metric space (X,d).
negative_passages.docid: 95831
negative_passages.text: Compactness and sequential compactness in metric spaces
negative_passages.docid: 143674
negative_passages.text: Space of bounded functions is complete
negative_passages.docid: 138607
negative_passages.text: Proof that a limit point compact metric space is compact.
negative_passages.docid: 35652
negative_passages.text: Every compact metric space is complete
negative_passages.docid: 43376
negative_passages.text: If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved