nLab subnet

Idea

The definition of the concept of sub-nets of a net requires some care. The point of the definition is to ensure that in classical mathematics, compact spaces are equivalently those for which every net has a converging subnet.

Definition

There are several different definitions of ‘subnet’ in the literature, all of which intend to generalise the concept of subsequences. We state them now in order of increasing generality. Note that it is Definition which is correct in that it corresponds precisely to refinement of filters. However, the other two definitions (def. , def. ) are sufficient (in a sense made precise by theorem below) and may be easier to work with.

Definition

(Willard, 1970).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:

We have a function $f\colon B \to A$ such that

• $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
• $f$ is monotone (that is, for every $\beta_1 \geq \beta_2 \in B$, $f(\beta_1) \geq f(\beta_2)$);
• $f$ is cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that $f(\beta) \geq \alpha$).

Definition

(Kelley, 1955).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:

We have a function $f\colon B \to A$ such that

• $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
• $f$ is strongly cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$, $f(\beta_1) \geq \alpha$).

Remark

Notice that the function $f$ in definitions and is not required to be an injection, and it need not be. As a result, a sequence regarded as a net in general has more sub-nets than it has sub-sequences.

Definition

(Smiley, 1957; Årnes & Andenæs, 1972).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a subnet of $x$ if:

The eventuality filter of $y$ refines the eventuality filter of $x$. (Explicitly, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$ there is an $\alpha_1 \geq \alpha \in A$ such that $y_{\beta_1} = x_{\alpha_1}$.)

The equivalence between these definitions is as follows:

Theorem

(Schechter, 1996).

1. If $y$ is a ()-subnet of $x$, then $y$ is also a ()-subnet of $x$, using the same function $f$.
2. If $y$ is a ()-subnet of $x$, then $y$ is also a ()-subnet of $x$.
3. If $y$ is a ()-subnet of $x$, then there is some net $z$ such that
   • $z$ is equivalent to $y$ in the sense that $y$ and $z$ are ()-subnets of each other, and
   • $z$ is a ()-subnet of $x$, using some function.

So from the perspective of definition (), there are enough ()-subnets and ()-subnets, up to equivalence.

References

A textbook account is in

• Eric Schechter, sections 7.14–7.21 of Handbook of Analysis and its Foundations, Academic Press (1996)

Lecture notes include

• Stijn Vermeeren, Sequences and nets in topology, 2010 (pdf)