Let (X,T) be a topological space,
let C be a closed subset of X,
let U be an open subset of X.
Prove that C - U is closed and U - C is open.
The Attempt at a Solution
I was trying to do this by 4 cases:
Case 1: Let U be a proper subset of C.
Then U - C = empty and hence is open.
Case 2: Let C be a proper subset of U.
Then C - U is empty and hence is closed.
Case 3: Let U be a proper subset of C.
Then C - U = ?
Case 4: Let C be a proper subset of U.
Then U - C = ?
Cases 3 and 4 are where I am stuck. Can someone give me some intuition to get me started on them? Thank you!