Proof That the Cartesian Product of Closed Sets is Closed
Theorem
If
foreach
then
Proof
Let
Let
where
isopen in
then
where
isthe part of
whereeachisexceptfor the
th,which is
Wethen have
Hence
and
Theorem
Ifforeachthen
Proof
Let
Letwhereisopen inthenwhereisthe part ofwhereeachisexceptfor theth,which isWethen have
Henceand