A Structure Sheaf for Kirch Topology

Kirch topology on $\mathbb{N}$ goes back to 1969, and is remarkable for being Hausdorff, connected, and locally connected. In this sense, it is analogous to the usual topology on $\mathbb{C},$ yet, to the author’s knowledge, there have been no Kirch topology analogs of the sheaf of complex-analytic functions until very recently. In our latest paper we constructed such natural sheaf of rings, the sheaf of locally LIP functions. In this paper we investigate some of its basic properties, primarily regarding zeroth and first cohomology and Cech cohomology with respect to covers by basic open sets.