6

I'm looking for an example of a topological space $X$, a sequence $(x_n)_{n \in \mathbb{N}}$ in $X$ and a converging subnet $(x_i)_{i\in I}$ of $(x_n)$, but with the property that $x_n$ does not have any converging subsequence.

I have an examples of $X$ and $(x_n)$ such that there is converging subnets $(x_i)$ and no converging subsequence of $(x_n)$, but I like to a explicit example of such a subnet $(x_i)$. (I still can't imagine how such subnets could look like...).

Thank you very much in advance

user110894
  • 63

1 Answers1

6

Let $X=\beta\Bbb N$, the Čech-Stone compactification of $\Bbb N$. Let $\mathscr{U}\in(\beta\Bbb N)\setminus\Bbb N$. Let

$$D=\{\langle n,U\rangle\in\Bbb N\times\mathscr{U}:n\in U\}\;,$$

and for $\langle m,U\rangle,\langle n,V\rangle\in D$ write $\langle m,U\rangle\preceq\langle n,V\rangle$ iff $U\supseteq V$; $\langle D,\preceq\rangle$ is a directed set. The net

$$\nu:D\to\Bbb N:\langle n,U\rangle\mapsto n$$

converges to $\mathscr{U}$ in $\beta\Bbb N$. Let $\sigma=\langle n:n\in\Bbb N\rangle$; the sequence $\sigma$ has no convergent subsequence in $\beta\Bbb N$. However, $\nu$ is a subnet of $\sigma$: for each $A\subseteq\Bbb N$, if $\sigma$ is eventually in $A$, then so is $\nu$.

(Note that I use the definition of subnet from J.F. Aarnes & P.R. Andenæs, ‘On nets and filters’, Math. Scand. $31 (1972)$, $285$-$282$ and these excellent notes by Saitulaa Naranong, not the slightly inferior one found, for example, in Kelly.)

Brian M. Scott
  • 631,399