X is infinite set with the finite complement topology. X is Hausdorff ??? why??? please thank...

X is infinite set with the finite complement topology.

X is Hausdorff ??? why???

please thank U

Expert Solution

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.

Please rate my answer and comment for doubts.

orchestra answered 1 year ago

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