Theorem
Ifis a T2 space then the set
is closed.
Proof
Forthere are open sets
and
such that
and
Henceand so
hence the diagonal of a T2 space is closed. The converse is also true, so that if the diagonal of
is closed, then
is a T2 space.
Theorem
Ifis a T2 space then the set
is closed.
Proof
Forthere are open sets
and
such that
and
Henceand so
hence the diagonal of a T2 space is closed. The converse is also true, so that if the diagonal of
is closed, then
is a T2 space.