Standard definition of subnets.

Question

Reading Willard's General Topology, I found the following definition of a subnet of a net $f:D\to X$:

$g:E\to X$ is a subnet of $f$ if there exists an increasing $\varphi:E\to D$ such that $g=f\circ\varphi$ and $\forall \alpha\in D, \exists \beta \in E, \varphi(\beta)\ge\alpha$;

while, reading Kelley's General Topology, I found the following definition of a subnet of net $f:D\to X$:

$g:E\to X$ is a subnet of $f$ if there exists $\varphi:E\to D$ such that $g=f\circ\varphi$ and$\forall \alpha\in D, \exists \beta_0 \in E, \forall\beta \ge\beta_0, \varphi(\beta)\ge\alpha.$

Clearly, Willard's definition is strictly stronger than Kelley's definition (i.e. every subnet according Willard is a subnet according Kelley but in general not the converse). Obviously we can use both definitions to obtain the standard theorems about subnets we were looking for, so it seems just a matter of tastes which definition we decide to use. However:

In the literature, which one of the two definitions has become standard? Willard's or Kelley's?

Thx for the comment... at this point I see that I have problems in searching topics in this site... — Bob, Mar 26 '19 at 07:55
For what it’s worth, so do I. I Googled it instead, found this question there. I think it’s very difficult to search this site. — MPW, Mar 26 '19 at 07:57

score 3 · Accepted Answer · answered Mar 26 '19 at 07:58

3

This is indeed mostly a matter of taste; Willard's definition is more in the style of subsequences (maybe easier to visualise), while Kelley's is more general (so easier to fulfil, which can be easier to construct in proofs e.g.) and more in the generality spirit of directed sets, as it were. A subnet in Kelley's sense need not be one in Willard's sense, but the difference is quiet subtle. In terms of translating to filters a third notion (AA-subnet) is also sometimes used. See the discussion in this thread e.g. Most papers I've seen tend to use filters or use Kelley's definition (but I haven't done a count of popularity).

answered Mar 26 '19 at 07:58

Henno Brandsma

242,131

A subnet of the net f from D into X is f restricted to a cofinal subset of D. Is not that another equivalent definition? – William Elliot Mar 26 '19 at 08:12
@WilliamElliot no, that's much too restrictive; you don't get the theorem that a space is compact iff every net has a convergent subnet, which is one of the main reasons for defining subnet in the first place. A sequence in your proposal only subnets that are subsequences while there are quite a few compact but not sequentially compact spaces... – Henno Brandsma Mar 26 '19 at 13:11

Standard definition of subnets.

1 Answers1