User:Kompik/Math/Subnet

Various definitions of subnet
See Talk:Subnet (mathematics) and Talk:Cofinal (mathematics) to know what's going on here. In order to compare the two different definitions of a subnet known in mathematical literature I have looked this definition up in several topological books I have and I have also tried to add some references using Google Book Search. Feel free to add definitions from other books.

"Kelley's" definition (or an equivalent one): [AB, Definition 2.15], [BR, p.206], [Dud, p.48], [En], [Fo], [Ke, p.70], [Pe, 1.3.2], [Ru, Definition 3.3.14].

[Ga, p.80] contains still another definition using the notion of the filter of upper sections of directed sets. Again, this is equivalent to Kelley's definition.

The notion of cofinal map is mentioned explicitly only in [Ru, Definition 3.3.14].

"Willard's" definition (or an equivalent one):

[Be, p.149], [Ge, p.119], [Mu, p.188], [Wi, Definition 11.2]

The notion of cofinal map is mentioned explicitly only in [Wi].

I haven't found a book or paper containing both definitions and comparing them somehow.