Prove that the Set C\D is closed if C is closed and D is open.

by Daniel   Last Updated January 12, 2018 20:20 PM

Suppose we have two sets in R^n, C and D, with C being closed and D being open. We are asked to prove that the set C\D is closed.

However, if we consider the complement of D\C, we get the intersection of C and the complement of D. But, since D is open, it must be that the complement of D is closed by definition. So, the complement of D\C is simply the intersection of two closed sets, so it must be closed. We then have that D\C must be open.

There clearly must be an error in my logic. Could someone hint me at where I'm going wrong?

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