Nets/Moore-Smith Sequences
Sequences are common objects in the field of topology. Often, sequences can help identify continuous functions, limit points and compact spaces in metric spaces.
Moore-Smith sequences(or nets) are essentially a generalization of the sequence for an arbitrary topological space and we can see that many foundational theorems of general topology can be stated in terms of nets.
So first, let’s recall what a partial order and direct set is:
A partial order is an order realtion satisfying:-
- If
and
, then
.
- If
, then
.
A directed set is a set with a partial order relation
such that for any
,there exists
such that
.
It’s best to think of a directed set as a sort of analogue to an indexing set.
A subset of a directed set
is said to be ‘frequently in’/cofinal in
if for each
, there exists some
such that
.
A good way(maybe not the best) to intuitively think of this is to imagine the directed set as a line where ‘going forward’, the points oscillate unevenly along the line and a subset is said to be cofinal if the points will occasionally enter into the subset. Of course, such an intuition is slightly flawed because the set must have a partial order and that isn’t exactly taken into account.
Now, that these terms are defined, we can define a Moore-Smith sequence or net.
A net in a topological space is a function
where
is a directed set.
is an element of
where
.
To simplify notation and tie the notion of a net back to that of a normal sequence, one can replace by
. So, a net in
can be represented by
.
If is a subset of
, then a net
is eventually in
if there exists some
such that for every
such that
.
Similarly, convergence can also be defined on a net. A net in is said to converge to a point
of
if for every neighborhood
of
,
is eventually in
.
Reforming the definition of continuity
In metric spaces, one definition for the continuity of a function is that for every sequence
that converges to a point
converges to
.
But such a definition is termed as sequential continuity in general topology and is in fact, strictly weaker than the topological definition(except for first-countable spaces).
However, using nets, the definition becomes the same.
A function between two topological spaces
is continuous at a point
if and only if for every net
which convergences to
also converges to
.
Just like in metric spaces, one can define an accumulation point of a net. A point of a topological space
is said to be an accumulation point if for every neighborhood
of
, the net is frequently in
.
Subnets
Obviously, this is an analogue to the sub-sequences in metric spaces.
Let be a directed set and
be a net in
. If
is another directed set and
is a function which satisfies:
, the image set, is frequently in
.
Then, the composite function is called a subnet of
.
Often, the subnet is represented as .
.
This definition may seem rather convoluted but it is necessary to define this new function because here we’re dealing with an arbitrary directed set and not the natural numbers. Also, notice that the indexing sets of
are different but the function
‘translates’ the indices of
into
. The monotone condition is clearly necessary and so is the condition that the image set of
must be cofinal(draw a comparison with a normal subsequence where
).
Limit points and Compact Spaces
When it comes to determining whether a point is an accumulation point or a space is compact, the real force of the definition of a net can be seen where the theorems are direct analogues to the basic theorems of compactness and limit points that one finds in real analysis.
Theorem
Let be a topological space.
is an accumulation of the net
if and only if there some subnet of
converges to
.
Proof:
Assume that is a accumulation point of the net
. Let
be the set of all pairs
such that
is a neighborhood of
and
is in
.
Define a function which maps
to
. Giving
the reverse inclusion partial order where
if
,
becomes a directed set, the image set
becomes cofinal in
and
is a monotone function. So, the subnet
defined by
converges to
.
Proving the converse is trivial.
Similarly, even the definition of compactness can be presented in terms of nets as follows:
Theorem
is compact if and only if every net in
has an accumulation point.
I will not present the compelte proof here. Proving basically follows by defining
for a net
and using compactness to show that
has non-finite intersection and this set is clearly the set of accumulation points of the net
.
To prove the converse, you assume that there is no finite subcover for the space , then consider the collection
of finite subsets of the indexing set of the open cover. Make this set a directed set by imposing inclusion. Define a net with directed set
a map every element of
to an element of
which isn’t in any any of the open sets of the cover associated with the element of
. Clearly, such a net can’t have an accumulation point.
If you noticed, the definition is an obvious generalization of the Bolzano-Weirestrass theorem. I have one last remark about nets.
A universal net/ultra-net is a net on a topological space which is eventually in every subset of
.
This concept combined with Zorn’s Lemma turns out to be rather interesting in providing an extremely simple proof of the famous Tychonoff theorem which states that an arbitrary product of compact spaces is compact. You can find the proof here.