Let be a mapping. We will prove that
, with equality when
is injective. Note that
does not have to be closed, open, or even continuous for this to be true. It can be any mapping.
Let . The mapping of
in
is
. As for
, it may overlap with
, we the mapping be not be injective. Hence,
.
>Taking on both sides, we get
.
How can we take the inverse on both sides and determine this fact? Is the reasoning valid? Yes. All the points in that map to
also map to
. However, there may be some points in
that do not map to
.
Are there other analogous points about mappings in general? In , select two sets
and
such that
. Then