Proof That a Function is Continuous if and Only if the Inverse Image of a Closed Set is Closed
Theorem
A function
is continuous if and only if the inverse image of every closed subset of
is closed in
Proof
Ifis continuous then
Supposeis continuous and
is any closed subset of
then
is an open subset ofand
is an open subset of
Suppose for every
is closed inand
is closed in
Let
be an open subset ofthen
is closed inand
is closed in
Hence
is an open subset of
and
is continuous.