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Compact space: Quiz

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Question 1:
Which of the following titles did Compact space have?
Exhaustion by compact sets
Super Robot Wars Compact 2
Sport Compact Car
Countably compact

Question 2: The spectrum of any commutative ring with the ________ (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases).
Zariski topologyField (mathematics)Algebraic geometryAlgebraic variety

Question 3: Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general ________.
Metric spaceTopological spaceCompact spaceTopology

Question 4: Thus, while disks and ________ are compact, infinite lines and planes are not, nor is a disk or a sphere with a missing point.
ManifoldGeometrySphere3-sphere

Question 5: Formally, a ________ is called compact if each of its open covers has a finite subcover.
Topological spaceHausdorff spaceMetric spaceCompact space

Question 6: The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by ________.
Hermann SchwarzBernard BolzanoKarl WeierstrassGeorg Cantor

Question 7: A metric space is called pre-compact or ________ if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces.
Compact spaceTotally bounded spaceComplete metric spaceHilbert space

Question 8: A metric space (or more generally any first-countable uniform space) is compact if and only if every ________ in the space has a convergent subsequence.
SequenceVector spaceMathematicsPartially ordered set

Question 9: (Gelfand–Naimark theorem) Properties of the ________ of continuous functions on a compact Hausdorff space are central to abstract analysis.
Banach spaceVector spaceHilbert spaceComplete metric space

Question 10: A subset of ________ in particular is called compact if it is closed and bounded.
Euclidean spaceMathematicsDimensionVector space







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