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 .
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.
Let be a topological space. is an accumulation of the net if and only if there some subnet of converges to .
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:
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.