Abstract
In this paper we examine properties of increasing sequence of Kirch-type topologies Dm defined on the set of positive integers that are subtopologies of Golomb's topology D. We give the formula for calculating the closures of arithmetic progressions and characterize regular open arithmetic progressions in all Kirch-type topologies Dm. Moreover, we examine which of spaces (N,Dm) are semiregular. Finally, for each m∈N we present conditions which are equivalent to continuity of non-constant polynomials f:(N,Dm)→(N,Dm).
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