Convergence of funcoids * by Victor Porton March 30, 2010 Abstract Considered convergence and limit for funcoids (a generalization of proximity spaces). I also have defined (generalized) limit for arbitrary (not necessarily continuous) func- tions under certain conditions. This article is a part of my Algebraic General Topology research. A.M.S. subject classification: 54A20, 54E05 Table of contents 1 Draft status ................................................. ? 2 Common ................................................... ? 3 Convergence ................................................. ? 4 Limit ...................................................... ? 5 Generalized limit ............................................. ? 5.1 The definition .............................................. ? 5.2 Generalized limits as a generalization of limits ......................... ? 5.3 Yet to do ................................................. ? Bibliography .................................................. ? 1 Draft status This is a partial draft. 2 Common See [2] for the definition of funcoid. 3 Convergence Definition 1. A filter object F converges to a filter object A regarding a funcoid µ (F→ μ A) iff F⊆〈 µ 〉A. 1 Definition 2. A funcoid f converges to a filter object A regarding a funcoid µ (f → μ A) iff im f ⊆〈 µ 〉A that is iff im f → μ A. Definition 3. A funcoid f converges to a filter object A on a filter object B regarding a funcoid µ iff f | B → μ A. *. This document has been written using the GNU T E X MACS text editor (see www.texmacs.org). 1. This generalizes the standard definition of filter convergent to a point or to a set. 1