limit points of Aintersect B, closure of (AintersectB)
I was given a problem from my book.
a) If y is a limit point of AUB, show that y is either a limit point of A or a limit point of B.
b) Prove that (AUB) closure=A closureUBclosure
c) Does the result about closures in b)extend to infinite unions in sets
I was basically asked to consider this problems with intersections instead of unions.
a)Let y be a limit pt of AintersectB
y_n is an element of A intersect B
So some subsequence (y_n) must lie in A and B
So y is a limit pt of A and B
b)I will consider (AintersectB)closure=Aclosure intersect Bclosure and want to see if this is true.
Part a says L(AintersectB)=L(A)intersectL(B)
So (AintersectB)closure=A intersect BU L(A) intersect L(B)
=(AUL(A)) intersect (BintersectL(B)
=Aclosure intersect B intersect L(B)
I'm not entirely sure if this is correct
c) I'm not sure how to get started