Continuous function
Let and
be two topological spaces and
a function.The function f is said continuous in
if and only if for all V a neighborhood of f(x) we have
is a neighborhood of x. f is continuous on X if and only if f is continuous in any
.
Theorem
Let and
be two topological spaces and
a function. The following statement are equivalent
(i) f is continuous on X;
(ii) we have
;
(iii) closed in
we have
is closed in
;
(iv) we have
;
(v) we have
Proof.
Let
.
means that
is open in
. But a set is open if and only if the set is a neighborhood for any of its points.
So, let then
but
is open in
so
and as f is continuous in x we have
and so
is a neighborhood for any of its points, so it is open.
Let F be closed in Y. Then
is open in Y and by (ii)
is open in X, so
is closed in X.
Let be
. As
we have
. But from (iii) as
is closed in
,
is closed in
and if a set
is included in a closed set, its closure is equally included in the same closed set, so
;
Let
, then
=
or
Let
and let
, then
or
and so
so f is continuous in an arbitrary
so f is continuous on
.