In: Statistics and Probability
No, X is not Hausdorff.
(A)We’ll prove that no two open sets are disjoint, which implies
the result. BWOC, let
U, V be disjoint open sets. This implies
V ⊂
, and
is a finite set, which implies V
is finite. But then
is infinite, contradicting the fact that V is open.
(B) Let A ⊂ X and let
{U}
be an open cover. Pick any single element of the cover, U∗
. It’s complement has finitely many
elements, so there are only finitely many elements of A, x1, . . .
, xn, that are not in this set. For each
one, find a Ui
containing xi. Then U∗
,{Uαi}is
a finite subcover.
(C) Any open set other than X
or ∅ is a set that is compact, but not closed.
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