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Let's define a net and subnet in this way:

  • A net is any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set.
  • A net $m:(P',\le)\to X$ is a subnet of the net $n:(P,\le)\to X$ iff there is a function: $$\theta:(P',\le)\to (P,\le)$$ which is increasing: $$x'\le y' \to \theta(x')\le\theta (y')$$ and cofinal: $$(\forall p\in P)(\exists p'\in P')(p\le\theta(p'))$$ and $m=n\circ\theta$.
  • The filter assigned to a net $n:(P\le)\to X$ is the filter generated by $$\lbrace \lbrace n(x)\mid x\ge p \rbrace \mid p\in P\rbrace$$ on X. We denote this filter by $\mathcal F_n$,

My question is:

Are there nets $m:(P',\le)\to X$ and $n:(P,\le)\to X$ with $$\mathcal F_n\subseteq \mathcal F_m$$ such that $m$ is not a subnet of $n$?

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Yes. Let Z be the integers, let X consist of a single point and let m : {0} -> X and n : Z -> X be constant functions. Then n and m give the same filter but m cannot be a subnet of n since no single integer in Z is cofinal.

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