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(Un-)Countable union of open sets - Mathematics Stack Exchange A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in this case, the union or intersection of any finite collection of open sets is open) the validity of the property for an infinite collection doesn't follow from that In other words, induction helps you prove a
For what $n$ is $U_n$ cyclic? - Mathematics Stack Exchange 17 Un U n is cyclic iff n n is 2 2, 4 4, pk p k, or 2pk 2 p k, where p p is an odd prime The proof follows from the Chinese Remainder Theorem for rings and the fact that Cm ×Cn C m × C n is cyclic iff (m,n)= 1 (m, n) = 1 (here Cn C n is the cyclic group of order n n)
geometry - Circle revolutions rolling around another circle . . . At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this? Consider Circle A rolling along a straight line the length of the circumference of Circle B Then it will revolve 3 times