query: Let $(X,\mathscr T)$ be a metrizable space such that every metric that generates $\mathscr T$ is bounded. Prove that $X$ is compact.
positive_passages.docid: 143484
positive_passages.text: A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
negative_passages.docid: 28637
negative_passages.text: Proving continuous image of compact sets are compact
negative_passages.docid: 17165
negative_passages.text: Is total boundedness a topological property?
negative_passages.docid: 47903
negative_passages.text: How to prove every closed interval in R is compact?
negative_passages.docid: 69499
negative_passages.text: Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
negative_passages.docid: 47810
negative_passages.text: $C(X)$ is separable when $X$ is compact?
negative_passages.docid: 137067
negative_passages.text: For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.
negative_passages.docid: 143674
negative_passages.text: Space of bounded functions is complete
negative_passages.docid: 154978
negative_passages.text: Some equivalent formulations of compactness of a metric space
negative_passages.docid: 19111
negative_passages.text: If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 3708
negative_passages.text: How can I prove that compact subset iff sequentially compact in second countable topological spaces?
negative_passages.docid: 156304
negative_passages.text: Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?
negative_passages.docid: 14509
negative_passages.text: Contraction Map on Compact Normed Space has a Fixed Point
negative_passages.docid: 149220
negative_passages.text: If every compact set is closed, then is the space Hausdorff?
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 9268
negative_passages.text: Proving that the Union of Two Compact Sets is Compact