Continuity redux
So now we have two new ways to talk about topologies: neighborhoods, and closure operators. We can turn around and talk about continuity directly in our new languages, rather than translating them into the open set definition we started with.
First let’s tackle neighborhoods. Remember that a continuous function from a topological space
to
is one which pulls back open sets. That is, to every open set
there is an open set
which
sends into
. But in the neighborhood definition we don’t have open sets at the beginning; we just have neighborhoods of points.
What we do is notice that a neighborhood of a point
is a set which contains an open set
containing
. In particular we can consider neighborhoods of a point
. The the preimage
is an open set containing
, which is a neighborhood! So, given a neighborhood
of
there is a neighborhood
of
so that
. This is an implication of the definition of continuity written in the language of neighborhoods, and it turns out that we can turn around and derive our definition of continuity from this condition.
To this end, we consider sets and
with neighborhood systems
and
, respectively. We will say that a function
is continuous at
if for every neighborhood
there is a neighborhood
so that
, and that
is continuous if it is continuous at each point in
.
Now, let be an open set in
. That is, a set which is a neighborhood of each of its points. We must now show that
is a neighborhood of each of its points. So consider such a point
, and its image
Since we are assuming that
is a neighborhood of
, there must be a neighborhood
of
so that
. But then
, and since the neighborhoods of
form a filter this means
is a neighborhood as well. Thus the preimage of an open set is open.
In particular, we can consider a set and its interior
, which is an open set contained in
. And so its preimage
is an open set contained in
. Thus we see that
. Finally, we can dualize this property to see that
. That is, the image of the closure of
is contained in the closure of the image of
for all subsets
. Let’s now take this as our definition of continuity, and derive the original definition from it.
Well, first let’s just dualize this condition to get back to say that for all sets
. Now any open set
is its own interior, so
. But
by the definition of the interior. And so
is its own interior, and is thus open.