- #1
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Homework Statement
Again, this one seems suspiciously easy, so I'd like to check if I'm missing something.
One needs to show that X is completely normal iff for every pair of separated sets A, B in X (i.e. sets such that both Cl(A) and B, and A and Cl(B) are disjoint), there exist disjoint open sets containing them.
The Attempt at a Solution
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Let S be a subspace of X, and take disjoint closed subsets A, B of X. It follows that they are separated, so from the hypothesis it follows that they have disjoint open neighborhoods containing them. Hence S is normal.
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Let A and B be two separated subsets of X. Since X\Cl(B) is open, and contains all of A, for any a in A there is a neighborhood U of a disjoint from B. A is contained in the union of these neighborhoods, and this union is open and disjoint from B, and contains A. Do this for the set B, and you get an open set containing B and disjoint from A.
But I smell this is not true, since I didn't use the fact that X is completely normal here.