A game characterization of limit-detecting sequences in locally compact $G$-spaces
Тарас Онуфриевич Банах, S. I. Pidkuyko · Matematychni Studii · 2004
Abstract. A sequence S = {xn}n∈ω in a locally compact G-space X is called (strongly) limit-detecting if a continuous function f: X → R has limit limx→ ∞ f(x) provided for any g ∈ G (any g from a given neighborhood of the unit of G) the limit limn→ ∞ f(gxn) exists; S is called κ-controlling for a cardinal κ if for any collection U of open unbounded subsets of X with |U | = κ there is g ∈ G such that for any U ∈ U the intersection U ∩ gS is unbounded. It is proved that under some mild conditions a set S ⊂ X is strongly limit-detecting if and only if S is ω-controlling in X if and only if S is asymptotically dense in the sense that for any neighborhood U of the unit in G the set US has bounded complement in X. On the other hand, S ⊂ X is limit-detecting if and only if S is 1-controlling and splittable (which means that for any disjoint unbounded subsets U, V ⊂ X whose union U ∪ V has bounded complement in X there is g ∈ G such that both the intersections gS ∩ U, gS ∩ V are unbounded in X). In its turn, a set S ⊂ X is 1-controlling if the product KS is ω-controlling for some compact countable set K ⊂ G. These results are proved with help of some infinite game resembling the Telgárski game characterizing K-scattered properties. The textbook [4] of problems in Mathematical Analysis contains the following Problem