Standard definition of subnets.

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?

• You may be interested in this question – MPW Mar 26 at 7:43
• Thx for the comment... at this point I see that I have problems in searching topics in this site... – Bob Mar 26 at 7:55
• 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 at 7:57