Theorem
Ifis T4 and
is a closed subset of
then a continuous function
can be extended to a continuous function
Proof
is a continuous function of
mapping any element of
to an element of
Hence it can be written as
Each of the functionsis continuous and maps
into
Tietze's extension theorem tells us that each of the functions gives rise to a continuous function
from
into
Define
Thenis continuous and is a continuous extension of
from
to