query: If $h:X \to Y$ is a homeomorphism of metric spaces, how do I prove that $X$ is compact if and only if $Y$ is compact?
positive_passages.docid: 19566
positive_passages.text: Continuous images of compact sets are compact
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: 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: 83166
negative_passages.text: Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem
negative_passages.docid: 84004
negative_passages.text: Prove there is no contraction mapping from compact metric space onto itself
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 6118
negative_passages.text: Homeomorphism of the real line-Topology
negative_passages.docid: 37092
negative_passages.text: Homeomorphism between a cone in a compact space and this compactification
negative_passages.docid: 103397
negative_passages.text: Homeomorphisms between infinite-dimensional Banach spaces and their spheres
negative_passages.docid: 128069
negative_passages.text: Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
negative_passages.docid: 118851
negative_passages.text: $X$ be a Hausdorff topological space , let $f:X \to X$ be a continuous map such that $f\circ f=f$ , then is $f(X)$ is closed in $X$ ?
negative_passages.docid: 152493
negative_passages.text: Is an expanding map on a compact metric space continuous?
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: 121135
negative_passages.text: Why don't we use closed covers to define compactness of metric space?
negative_passages.docid: 106256
negative_passages.text: Continuous mapping on a compact metric space is uniformly continuous
negative_passages.docid: 113534
negative_passages.text: Show that any metrizable space $X$ is Hausdorff
negative_passages.docid: 18040
negative_passages.text: Show that a compact metric space is bounded