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

Then y=limy_n

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.

(AintersectB)closure=AintersectBUL(AintersectB)

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