Abstract

ALBERTO-DOMINGUEZ, José del Carmen; ACOSTA, Gerardo; DELGADILLO-PINON, Gerardo  and  MADRIZ-MENDOZA, Maira. The Golomb space and its non connectedness “im kleinen”. Integración - UIS [online]. 2017, vol.35, n.2, pp.189-213. ISSN 0120-419X.  https://doi.org/10.18273/revint.v35n2-2017004.

In the present paper we study Brown spaces which are connected and not completely Hausdorff. Using arithmetic progressions, we construct a base BG for a topology τG on N, and show that (N, τG), called the Golomb space is a Brown space. We also show that some elements of BG are Brown spaces, while others are totally separated. We write some consequences of such result. For example, the space (N, τG) is not connected “im kleinen” at each of its points. This generalizes a result proved by Kirch in 1969. We also present a simpler proof of a result given by Szczuka in 2010.

Keywords : Arithmetic progression; connectedness; connectedness “im kleinen”; Golomb topology; local connectedness.

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