Henri Cartan’s notion of a filter is equivalent to these constructs, which provide topological information about a space through an external indexing set. For 10 points each:
[10h] Name these generalizations of sequences that replace the natural numbers with an arbitrary directed set. These constructs are named for how they converge by “enclosing” on a point.
ANSWER: nets [accept Moore–Smith sequences]
[10e] Because nets are indexed by directed sets, one may define this operation to talk about convergence. L’Hopital’s rule computes this operation, which returns the value that a function approaches.
ANSWER: limits [accept limit points or accumulation points or cluster points]
[10m] A set A has this property if it contains the limit points of every net in A. Nets may be used to generalize the Heine–Borel theorem, which states that a bounded set with this property is compact.
ANSWER: closed [accept word forms; reject “clopen”]
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