I have three problems that I can't come up with examples for.
1.) Give an example of two sets, E1 and E2, such that closure(E1 intersect E2) does not equal closure(E1) intersect closure(E2)
2.) Give an example of two sets, E1 and E2, such that int(E1 U E2) does not equal int(E1) U int(E2)
3.) Give an example of a sequence of closed sets F1, F2, F3,... whose union is neither open nor closed.
I was told that any open set like (1,3) and a closed set [2,4] would work for number 1, but then closure(E1 intersect E2)=[2,3] and closure(E1) intersect closure(E2) = [2,3] so I do not know what to do.
I can't come up with anything else for any of them, so any help would be greatly appreciated.
1) try (0,2) and (2,4). Their intersection is { } so closure is empty set right? Then closure of (0,2) is [0,2] intersect closure (2,4) is [2,4] and you get an intersection of 2.
2)try int([0,1] u [1,2]) = int [0,2] = (0,2) and int[0,1] u int[1,2] = (0,1) u (1,2)