SEQUENTIAL ENDS AND NONSTANDARD INFINITE BOUNDARIES OF COARSE SPACES (Research Trends on General Topology and its Related Fields)

Abstract

This paper is an addendum to the author's previous paper ["A nonstandard invariant of coarse spaces, " The Graduate Journal of Mathematics, vol. 5, no. 1, pp. 1-8, 2020.]. Miller et al. [B. Miller, L. Stibich, and J. Moore, "An invariant of metric spaces under homologous equivalences, " Mathematics Exchange, vol. 7, no. 1, pp. 12-19, 2010.] introduced a functor σ: Coarse* → Sets, where Coarse* is the category of pointed coarse spaces and coarse maps. DeLyser et al. [M. DeLyser, B. LaBuz, and M. Tobash, "Sequential ends of metric spaces, " 2013, arXiv:1303.0711.] introduced a functor ε: Coarse. → Sets, and proved that ε coincides with σ on Metr* ( the full subcategory of metrisable spaces). Using techniques of nonstandard analysis, the author in ["A nonstandard invariant of coarse spaces, " The Graduate Journal of Mathematics, vol. 5, no. 1, pp. 1-8, 2020.] provided a functor ι: l ⊆ Coarse, → Sets, where l is an arbitrary small full subcategory, and a natural transformation ω: σ ↾ l ⇒ ι. The surjectivity of ω has been proved for all proper geodesic metrisable spaces, while the injectivity has remained open. In this note, we first pointed out that w is the composition of two natural transformations φ ↾ l : σ ↾ l ⇒ ε ↾ l and ω' : ε ↾ l ⇒ ι, and then show that ω' is injective for all spaces in l. As a corollary, ω is injective for all metrisable spaces in l. This partially answers some of the problems posed in ["A nonstandard invariant of coarse spaces, " The Graduate Journal of Mathematics, vol. 5, no. 1, pp. 1-8, 2020.].