I'm in analysis right now, and I'm being told that for open set A and closed set B, A\B is definitely open. I feel like I've found a counterexample where A\B is closed, but every solution I've found claims this is wrong. Here's my solution:
Let A = {1/n for n in N} (open as it does not contain its limit point 0). A = {1, 1/2, 1/3, ... }
Let B = {1/(n+1) for n in N}U{0} (closed as it does contain its limit point 0, alternatively its complement is clearly open). B = {1/2, 1/3, 1/4, ... , 0}
A\B = {1}. The complement of A\B is {-inf,1}U{1,inf}, both of which are open intervals in R, hence open sets, and the union of open sets is open. The complement of A\B is open <=> A\B is closed. QED
Where is my mistake? I think the disagreement may come from whether a single element set is closed in general, but clearly for any non-trivial {a}, its complement in R (-inf,a)U(a,inf) is clearly open, so wouldnt it necessarily be closed?
Let A = {1/n for n in N} (open as it does not contain its limit point 0). A = {1, 1/2, 1/3, ... }
Let B = {1/(n+1) for n in N}U{0} (closed as it does contain its limit point 0, alternatively its complement is clearly open). B = {1/2, 1/3, 1/4, ... , 0}
A\B = {1}. The complement of A\B is {-inf,1}U{1,inf}, both of which are open intervals in R, hence open sets, and the union of open sets is open. The complement of A\B is open <=> A\B is closed. QED
Where is my mistake? I think the disagreement may come from whether a single element set is closed in general, but clearly for any non-trivial {a}, its complement in R (-inf,a)U(a,inf) is clearly open, so wouldnt it necessarily be closed?
