Fix an offset $b \in \Bbb{Z}$ and define $U_c = \{cX^k + b : X \in \Bbb{Z}\}$. Let $B = \{U_c: c \in \Bbb{Z}\}$ form a basis for a topology on the set $\Bbb{Z}$. I have proved that it is indeed a basis for a topology. Was wondering if anyone has used it in a paper, and if so where?

  • The space is $T_0$ but not $T_1$.

  • $\{\varnothing\} \cup \{U_c\}_{c \in \Bbb{Z}}$ is a $\pi$-system.

A similar thing can be done with $U_c = \{k X^c + b : X \in \Bbb{Z}\}, \ c \geq 1$.

  • $\begingroup$ Is $k$ also fixed ahead of time? As to its properties, clearly it's not $T_1$, since $b$ is in every nonempty open set. $\endgroup$ – Noah Schweber Nov 22 '15 at 3:34
  • $\begingroup$ @NoahSchweber, yes $k$ is fixed. $\endgroup$ – I Said Roll Up n Smoke Adjoint Nov 23 '15 at 0:54
  • $\begingroup$ Some similar topologies were considered below: $\endgroup$ – Alex Ravsky Nov 24 '15 at 6:13
  • $\begingroup$ Solomon W. Golomb Arithmetica topologica In: (ed.): General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the symposium held in Prague in September 1961. Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, 1962. pp. 179--186. $\endgroup$ – Alex Ravsky Nov 24 '15 at 6:13
  • $\begingroup$ Solomon W. Golomb A Connected Topology for the Integers, The American Mathematical Monthly 10 (1959), 66:8. references $\endgroup$ – Alex Ravsky Nov 24 '15 at 6:14

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