Is there a common directed set that induces $2$ monotone cofinal maps to $2$ other directed sets?

Question

Recently, I have come across a result that the diameter of set is equal to that of its weak closure. A proof is straightforward if below result is true.

Let $E$ be a locally convex Hausdorff t.v.s., $X \subseteq E$, and $x,y \in \overline X$. Then there are nets $(x_d)_{d\in D}$ and $(y_t)_{t\in T}$ in $X$ such that $x_d \to x$ and $y_t \to t$. Then there subnets $(x_{\varphi(h)})_{h\in H}$ and $(y_{\phi(h)})_{h\in H}$ such that $x_{\varphi(h)}-y_{\phi(h)} \to x-y$.

Is this conjecture indeed true?

@HennoBrandsma It's best that $\phi = \psi$, but $\phi \neq \psi$ is still sufficient for my proof :v — Analyst, Feb 17 '22 at 11:56
$\phi: H \to D$ and $\psi: H \to T$ so they cannot be equal. — Henno Brandsma, Feb 17 '22 at 11:57

score 2 · Accepted Answer · answered Feb 17 '22 at 11:31

2

Try $H:=D \times T$ in the product direction (i.e. $(d,t) \le (d',t')$ iff $d \le d'$ and $t \le t'$) and the projections $\pi_D:H \to D$ and $\pi_T: H \to T$ as connecting maps. This is a common idea in net theory.

answered Feb 17 '22 at 11:31

Henno Brandsma

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Is there a common directed set that induces $2$ monotone cofinal maps to $2$ other directed sets?

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