But why is the set open? If it's the complement of A and A is open, shouldn't the complement of an open set always be closed?
You must understand that and is known as a relative complement.
That is the complement of A relative to B.
You see is an open set in the underlying space which may not be .
In the real numbers which is not open.
But why is the set open? If it's the complement of A and A is open, shouldn't the complement of an open set always be closed?
Yes, that's true. But "B\A" is NOT the "complement" of A, which is, by definition, the set all points not in A, whether they are in B or not. If and , then an open set. The [b]complement of A is , a closed set.