Moore-Smith Sequences/Nets in General Topology

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 \leq satisfying:-

  • \alpha \leq \alpha
  • If \alpha \leq \beta and \beta \leq \alpha, then \alpha=\beta.
  • If \alpha \leq \beta,\beta \leq \gamma, then \alpha \leq \gamma.

 

A directed set I is a set with a partial order relation \leq such that for any \alpha,\beta \in I,there exists \gamma such that \alpha \leq \gamma,\beta \leq \gamma.

It’s best to think of a directed set as a sort of analogue to an indexing set.

A subset I of a directed set J is said to be ‘frequently in’/cofinal in J if for each \alpha \in J, there exists some \beta \in I such that \alpha \leq \beta.

 

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 X is a function f:J \mapsto X where J is a directed set. f(\alpha) is an element of X where alpha \in J.

To simplify notation and tie the notion of a net back to that of a normal sequence, one can replace f(\alpha) by x_{\alpha}. So, a net in X can be represented by (x_{\alpha}).

 

If Y is a subset of X, then a net (x_{\alpha}) is eventually in Y if there exists some \alpha \in J such that for every \beta such that \alpha \leq \beta,x_{\beta} \in Y.

 

Similarly, convergence can also be defined on a net. A net in X is said to converge to a point x of X if for every neighborhood U of x, (x_{\alpha})_{\alpha \in J} is eventually in U.

Reforming the definition of continuity

In metric spaces, one definition for the continuity of a function f is that for every sequence (x_{n})_{n \in \mathbb{N}} that converges to a point c,(f(x_{n}))_{n \in \mathbb{N}} converges to f(c).

 

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 f:X \mapsto Y between two topological spaces X,Y is continuous at a point xif and only if for every net (x_{\alpha}) which convergences to x,(f(x_{\alpha})) also converges to f(x).

 

Just like in metric spaces, one can define an accumulation point of a net. A point x of a topological space X is said to be an accumulation point if for every neighborhood U of x, the net is frequently in U.

Subnets

Obviously, this is an analogue to the sub-sequences in metric spaces.

Let J be a directed set and f:J \mapsto X be a net in X. If K is another directed set and g:K \mapsto J is a function which satisfies:

  • a \leq b \Rightarrow g(a) \leq g(b)
  • g(K), the image set, is frequently in J.

Then, the composite function f \circ g:K \mapsto X is called a subnet of (x_{\alpha}).

Often, the subnet is represented as (y_{\beta}).

 

y_{\beta}=x_{g(\beta)}.

This definition may seem rather convoluted but it is necessary to define this new function g because here we’re dealing with an arbitrary directed set and not the natural numbers. Also, notice that the indexing sets of K,J are different but the function g ‘translates’ the indices of K into J. The monotone condition is clearly necessary and so is the condition that the image set of g must be cofinal(draw a comparison with a normal subsequence where J=\mathbb{Z}^{+}).

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

LetX be a topological space. x is an accumulation of the net (x_{\alpha}) if and only if there some subnet of (x_{\alpha}) converges to x.

Proof:

Assume that y is a accumulation point of the net (x_{\alpha})_{\alpha \in J}. Let K be the set of all pairs (U,\alpha) such that U is a neighborhood of y and f(\alpha)=x_{\alpha} is in U.

Define a function g:K \mapsto J which maps (U,\alpha) to \alpha. Giving K the reverse inclusion partial order where (U,\alpha) \leq (V,\beta) if \alpha \leq \beta,V \subset U, K becomes a directed set, the image set g(K) becomes cofinal in J and g is a monotone function. So, the subnet (y_{\beta}) defined by y_{\beta}=x_{g(\beta)} converges to y.

Proving the converse is trivial.


Similarly, even the definition of compactness can be presented in terms of nets as follows:

 

Theorem

X is compact if and only if every net in X has an accumulation point.

I will not present the compelte proof here. Proving \Rightarrow basically follows by defining E_{\alpha}=\{x_{\beta}|\alpha \leq \beta \} for a net (x_{\alpha})_{\alpha \in J} and using compactness to show that \bigcap_{\alpha \in J} cl(E_{\alpha}) has non-finite intersection and this set is clearly the set of accumulation points of the net (x_{\alpha}).

To prove the converse, you assume that there is no finite subcover for the space X, then consider the collection D 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 D a map every element of D to an element of X which isn’t in any any of the open sets of the cover associated with the element of D. 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 X which is eventually in every subset of X.

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.

 

 

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