Various definitions of subnet edit
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.
References edit
[AB] Charalambos D. Aliprantis and Kim C. Border. Infinite Dimensional Analysis, A Hitchhiker’s Guide. Springer, Berlin, 3rd edition, 2006.
[Be] G.Beer: Topologies on Closed and Closed Convex Sets
[BR] Gerard Buskes and Arnoud van Rooij: Topological Spaces: From Distance to Neighborhood
[Dud] R. M. Dudley. Real Analysis and Probabilty. Cambridge University Press, Cambridge, 2002.
[Dug] James Dugundji. Topology. Allyn and Bacon, Boston, 1966.
[En] R. Engelking. General Topology. PWN, Warsaw, 1977.
[Fo] G. Folland. Real analysis.. modern techniques and their applications (2ed., PAM, Wiley, 1999)
[Ga] Werner Gähler: Grundstrukturen der Analysis I, Akademie - Verlag, Beriln, 1977 (in German)
[Ge] Michael C. Gemignani: Elementary Topology, Dover Publications, 1990
[Ke] John L. Kelley. General Topology. Springer.
[Mu] James R. Munkres. Topology. Prentice Hall, Upper Saddle River, 2nd edition, 2000.
[Pe] Gert K. Pedersen. Analysis Now. Springer-Verlag, New York, 1989. Graduate texts in mathematics 118.
[Ru] Volker Runde. A Taste of Topology. New York, 2005. Universitext.
[Wi] S. Willard. General topology. Addison-Wesley, Massachussets, 1970.