Give an example of a sequence $x_1, x_2, . . .$ in $\mathbb Z$ that converges in the even-spaced topology but not in the discrete topology.

Question

Give an example of a sequence $x_1, x_2, . . .$ in $\mathbb Z$ that converges in the even-spaced topology but not in the discrete topology.

I know that the topology generated by the arithmetic progressions is called the evenly-spaced topology or Furstenberg’s topology and that in the discrete topology, $x_1, x_2,...$ converge to $x$ means that there exists $n_0$ such that, for $n$ greater than or equal to $n_0$, $x_n = x$, because {$x$} is open.

But I can't think of such a sequence that converges in the even-spaced topology but not in the discrete topology. Any examples please?

Answer 1

Let's try, for example, find a sequence $\{z_n\}_{n\in \mathbb{N}}$ that converges to $0$ in the even space topology $T_e$ but not the discrete $T_d$.

The only thing needed to prevent convergence in $T_d$ is that the sequence is not eventually constant. So now we need convergence in $T_e$. I.e., for any basic open set $A$ (arithmetic progression) containing $0$, there must exist an $N$ such that $x_n$ is in $A$ for all $n\geq N$.

To do this, let $p_i$ denote the i'th prime number (and for convenience we let 1 be prime :P). Let $z_n = 0$ for all n that are odd, and let $z_{2k} = p_1\cdot p_2 \cdot \dots \cdot p_k$ be the product of the first $k$ primes. It is clear that this sequence eventually lives in every arithmetic progression containing $0$, and thus converges to $0$.

(The first terms of our sequence look like $0, 1, 0, 2, 0, 6, 0, 30, 0, 210\dots$ And this probably also works without the extra $0$'s)

Answer 2

$\Bbb Z$ is a second countable space (there are only countably many arithmetic progressions) such that all non-empty open sets are infinite. In particular there are no isolated points. All basic open sets are also closed so it is a regular space. So Sierpiński's theorem (and Urysohn) imply that $\Bbb Z \simeq \Bbb Q$ in this topology ($\Bbb Q$ has the Euclidean topology of course).

So plenty of non-constant convergent sequences exist; as many as there are real numbers.