Compactification of a convergence space

Vinod Kumar
1979 Proceedings of the American Mathematical Society  
A characterization for the class of convergence spaces having the largest Hausdorff compactification is given, and regularity and A-Hausdorffness of modified Richardson compactification are discussed. Introduction. A convergence space, though a generalisation of a topological space, may behave quite differently from a topological space, e.g. unlike a Hausdorff topological space every Hausdorff convergence space has a Hausdorff compactification which, of course, need not be the largest one [6] .
more » ... e largest one [6] . The question when a Hausdorff convergence space has the largest Hausdorff compactification seems to have considerable importance. In [4] and [5], necessary and sufficient conditions for a Hausdorff convergence space (X, qx) to have the largest Hausdorff compactification are found; it is observed in [8] that the proof of the necessity part is not sound and shown that the largest Hausdorff compactification of (A', qx), whenever it exists, is given by the modified Richardson compactification (X*, qx,); using this we find the largest class of Hausdorff convergence spaces having the largest Hausdorff compactification. We also discuss when (X*, qx.) is regular for a regular Hausdorff convergence space (X, qx), and when for a topological Tychonoff space (X, qx), (X*, qx.) and (XX*, a^.), the topological modification of (X*, qx.) [7] , are homeomorphic to ßX, the topological Stone-Cech compactification of (X, qx). Definitions and notations. For definitions, not given here, the reader is asked to refer to [2] and [7] . We shall follow the notations of [8] . For a set X, FX denotes the set of all filters on X and PX the set of all subsets of X. Let u.f. denote ultrafilter. For x E X, x =* {A c A^x E /!} is the principal filter containing {x). A convergence structure (cs.) on X is a function qx from FX to PX satisfying the following conditions: (1) for x E X, x E qx(x); (2) for tp, 4> E FX, if q> c rp, then qx(<p) c qx(\p); (3) if x E qx(<p), then x E qx(<p n x). The pair (X, qx) is called a convergence space. If x E qx(<p), we say that tp is
doi:10.1090/s0002-9939-1979-0516474-0 fatcat:2fg2p2ag55butkyxyprunmlmka