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I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don't fill the edge, where nets and sequences starts to behave differently.

Also it would be nice to find complete survey on uncountable sums. They are defined with usage of nets but there some operations with sums which require additional analysis.

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Take a look at these notes by Pete L. Clark that seem to contain all you need. The classical textbook reference on nets is Kelley's book‌​. What is explained in section 1.3 of Pedersen's Analysis Now turned out to be quite sufficient for all my needs. – t.b. Jan 4 '12 at 19:08
The notes by Pete L. Clark are very good. The best reference book on nets is the Handbook of Analysis and its Foundations by Eric Schechter. It contains lots of important material you will not find in Kelley's book – Michael Greinecker Jan 4 '12 at 19:13
@Michael: out of curiosity and for lack of access to that book: what is this important material? – t.b. Jan 4 '12 at 19:17
I think that very good reference on convergence of nets is Engelking's General topology.\\ @t.b. I guess some of interesting things in Schechter's book could be comparison of three different definitions of subnets, I typed some definitions and results from this part here:… – Martin Sleziak Jan 4 '12 at 19:22
One of the things is a study of different notions of subnet employed in the literature. There is also a lot on the relation between nets and filters. Most of this material seems to be actually in the notes of Pete, including material missing from Kelley. The book is also awsome in general. :-) – Michael Greinecker Jan 4 '12 at 19:31
up vote 5 down vote accepted

I'm collecting the comments together into an answer:

There seems to be no question that Pete L. Clark's Notes on Convergence are a very thorough treatment of nets, filters and convergence. There are also a few exercises on summation (which always is absolute summation).

In my experience there's no more need for basics than what's contained in chapter 1 of Pedersen's Analysis Now.

Two classic references are J.L. Kelley's book General Topology and R. Engelking's General Topology.

Michael Greinecker and Martin Sleziak both recommend Eric Schechter's Handbook of Analysis and its Foundations for a detailed discussion of the various notions of subnets that can be found in the literature. Parts of this is contained in these notes and these by Martin Sleziak.

Furthermore Aarnes and Adenæs, On Nets and Filters, Math. Scand 31 (1972), 285–292; was recommended.

Basic results on nets are given in Chapter 2 of Megginson's book An introduction to Banach Space Theory, Springer GTM 183. This book also covers nets in topological groups (starting from p.154) and topological vectors spaces (starting from p.167). These aspects of nets might be interesting to you if you plan to use nets in functional analysis. From the above mentioned books both Pedersen and Schechter are oriented towards analysis, too.

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I'm posting this as a CW-answer so please free to add to it. – t.b. Jan 4 '12 at 20:04
A couple of my old sci.math posts might be of use. (24 January 2003) discusses the early history of nets and filters. A number of references are given, along with excerpts from several of the references. (18 April 2000) gives several references for the use of nets in analysis. – Dave L. Renfro Jan 4 '12 at 20:23

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