Ultracompanions of subsets of a group. (English) Zbl 1324.54044

Protasov, I.; Slobodianiuk, S.

Commentat. Math. Univ. Carol. 55, No. 2, 257-265 (2014).

Summary: Let \(G\) be a group, \(\beta G\) be the Stone-Čech compactification of \(G\) endowed with the structure of a right topological semigroup and \(G^*=\beta G\setminus G\). Given any subset \(A\) of \(G\) and \(p\in G^*\), we define the \(p\)-companion \(\Delta_p(A)= A^*\cap Gp\) of \(A\), and characterize the subsets with finite and discrete ultracompanions.

MSC:

54D35	Extensions of spaces (compactifications, supercompactifications, completions, etc.)
22A15	Structure of topological semigroups
20F69	Asymptotic properties of groups

Keywords:

Stone-Čech compactification; ultracompanion; sparse and discrete subsets of a group