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hw1-12.pdf
(from Continuous Functions)
There are several topology proofs here regarding continuity.
Real Analysis - 1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable
2- if An is a countable set for each n belong to N,then Un=1(to infinity) An is countable