Let $G$ be a filter finer than the filter $F$. Is it necessary that the net based on the filter $G$ is a subnet of the net based on $F$?
Suppose $G$ is a finer filter than $F$ in a topological space $X$.  Is the net base in $G$ a subnet of the net base in $F$?  I am using the definitions of General Topology of Willard:     Definition 12.15. If $(x_\lambda)$ is a net in $X$, the filter generated by the filter base $\mathscr C$ consisting of the sets $B_{\lambda_0}=\{x_\lambda; \lambda\ge\lambda_0\}$, $\lambda_0\in\Lambda$, is called the filter generated by $(x_\lambda)$.      Definition 12.16. If $\mathscr F$ is a filter on $X$, let $\Lambda_{\mathscr F}=\{(x,F); x\in F\in\mathscr F\}$.   Then $\Lambda_{\mathscr F}$ is directed by the relation $(x_1,F_1)\le(x_2,F_2)$ iff $F_2\subset F_1$, so the map   $P\colon \Lambda_{\mathscr F} \to X$ defined by $P(x,F)=x$ is a net in $X$. It is called the net based on $\mathscr F$.   The definition of subnet used in Willard is the second one from this post:   Thank you
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