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Connections between connected topological spaces on the set of positive integers

100%

Szczuka P.

Open Mathematics

2013

tom 11

nr 5

876-881

In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (ℕ, T) and (ℕ, T′).

The closures of arithmetic progressions in the common division topology on the set of positive integers

84%

Szczuka P.

Open Mathematics

2014

tom 12

nr 7

1008-1014

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

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2 Open Mathematics

2 Szczuka P.

1 2014

1 2013

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Connections between connected topological spaces on the set of positive integers

The closures of arithmetic progressions in the common division topology on the set of positive integers