# Subnet vs Cofinal subnet

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.

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 $$p$$.