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Product topology: Quiz

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Question 1: The product topology is also called the topology of pointwise convergence because of the following fact: a ________ (or net) in X converges if and only if all its projections to the spaces Xi converge.
Function (mathematics)SequenceVector spacePartially ordered set

Question 2: This is easy to show for finite products, while the general statement is equivalent to the ________.
Mathematical logicZermelo–Fraenkel set theorySet theoryAxiom of choice

Question 3: A product of ________ need not be locally compact.
Dense setBaire spaceTopological spaceLocally compact space

Question 4: In topology and related areas of mathematics, a product space is the ________ of a family of topological spaces equipped with a natural topology called the product topology.
Binary relationCardinal numberDirect productCartesian product

Question 5: An important theorem about the product topology is Tychonoff's theorem: any product of ________ is compact.
Hilbert spaceCompact spaceTopological spaceMetric space

Question 6: If follows from the above universal property that a map f : YX is continuous ________ fi = pi o f is continuous for all i in I.
If and only ifFirst-order logicLogical connectivePropositional calculus

Question 7: The ________ is equivalent to the statement that the product of a collection of non-empty sets is non-empty.
Zermelo–Fraenkel set theoryMathematical logicAxiom of choiceSet theory

Question 8: Every product of ________ is Hausdorff[1]
Hausdorff spaceTopological spaceT1 spaceNormal space

Question 9: A map that "locally looks like" a canonical projection F × UU is called a ________.
ManifoldPrincipal bundleDifferentiable manifoldFiber bundle







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