query: $f:X\rightarrow Y$ is a closed map, $f^{-1}(\{ y\}) $ is compact $\forall y\in Y$ then $X$ is compact.
positive_passages.docid: 156304
positive_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: 106957
negative_passages.text: Compact sets are closed?
negative_passages.docid: 136583
negative_passages.text: Infimum is a continuous function, compact set
negative_passages.docid: 130299
negative_passages.text: Prove that if a function $f: X\to Y$ continuous then its graph is closed
negative_passages.docid: 111246
negative_passages.text: product of two continuous proper maps
negative_passages.docid: 12293
negative_passages.text: Let X be a T1 space, and show that X is normal if and only if each neighbourhood of a closed set F contains the closure of some neighbourhood of F
negative_passages.docid: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 135157
negative_passages.text: Totally bounded, complete $\implies$ compact
negative_passages.docid: 149820
negative_passages.text: How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
negative_passages.docid: 46788
negative_passages.text: Compact open topology on space of continuous maps from a discrete space is a product
negative_passages.docid: 51914
negative_passages.text: Show that $S$ is compact.
negative_passages.docid: 93102
negative_passages.text: prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 142901
negative_passages.text: "The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$" and its compactness
negative_passages.docid: 50026
negative_passages.text: Product of compact and closed in topological group is closed
negative_passages.docid: 128069
negative_passages.text: Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
negative_passages.docid: 97316
negative_passages.text: Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X