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I was reading the wikipedia page on subnets, and it says:

A more natural definition of a subnet would be to require B to be a cofinal subset of A and that h be the identity map. This concept, known as a cofinal subnet, turns out to be inadequate. For example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.

I don't see why in the Tychonoff plank a cluster point of a net is not equal to a limit of a subnet.

Does anyone know why, or a reference where to find out why this is true? Thanks.

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1 Answer 1

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Let $D=\omega_1\times\omega$ ordered lexicographically: $\langle\alpha,m\rangle\preceq\langle\beta,n\rangle$ iff $\alpha<\beta$, or $\alpha=\beta$ and $m\le n$; this is clearly a directed set. Let $$X=(\omega_1+1)\times(\omega+1)=[0,\omega_1]\times[0,\omega]$$ with the product topology (where the ordinal spaces have their usual order topologies), and for convenience let $p=\langle\omega_1,\omega\rangle$.

The net $\nu:D\to X:\langle\alpha,n\rangle\mapsto\langle\alpha,n\rangle$ has $p$ as a cluster point. To see this, let $U$ be any open nbhd of $p$ in $X$; then there are $\alpha_0\in\omega_1$ and $n_0\in\omega$ such that $\langle\alpha,n\rangle\in U$ whenever $\alpha>\alpha_0$ and $n>n_0$. Let $\langle\beta,m\rangle\in D$ be arbitrary, and set $\alpha=\max\{\beta,\alpha_0+1\}$ and $n=\max\{m,n_0+1\}$; then $\langle\beta,m\rangle\preceq\langle\alpha,n\rangle$, and $\nu(\langle\alpha,n\rangle)=\langle\alpha,n\rangle\in U$.

However, no cofinal subnet of $\nu$ converges to $p$. Suppose that $C\subseteq D$ is cofinal in $D$. For each $n\in\omega$ let $C_n=C\cap(\omega_1\times\{n\})$; $|C|=\omega_1$, so there is some $m\in\omega$ such that $|C_m|=\omega_1$. Then $C_m$ is cofinal in $C$, but $\nu[C_m]=C_m$ is disjoint from the open nbhd $[0,\omega_1]\times[m+1,\omega]$ of $p$, so $\nu\upharpoonright C$ does not converge to $p$.

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