query: What does compactness actually mean
positive_passages.docid: 25752
positive_passages.text: What should be the intuition when working with compactness?
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: 116663
negative_passages.text: If f is a continuous map from X to Y with X limit point compact, does it follow that f(X) is limit point compact?
negative_passages.docid: 481
negative_passages.text: Can compact sets completey determine a topology?
negative_passages.docid: 116542
negative_passages.text: Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology]
negative_passages.docid: 118508
negative_passages.text: Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact?
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 146146
negative_passages.text: Proof that a closed subset of a compact space is a compact space
negative_passages.docid: 10513
negative_passages.text: What is the significance of the term "separable" in the context of countability properties?
negative_passages.docid: 158814
negative_passages.text: Two definitions of locally compact space
negative_passages.docid: 47251
negative_passages.text: Non-compactness of $\mathbb{R}$ with the cocountable topology
negative_passages.docid: 22854
negative_passages.text: Let $A , B\subseteq\mathbb{R}$. If $A$ is closed and $B$ is compact, is $A\cdot B$ closed?
negative_passages.docid: 156382
negative_passages.text: Why is Compactification restricted to Toroids, Calabi-Yau et al?
negative_passages.docid: 145415
negative_passages.text: Examples of almost compact spaces.
negative_passages.docid: 84492
negative_passages.text: Can $X$ have compact connected components?
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 72209
negative_passages.text: show that torus is compact