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## Homework Statement

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!