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True or false?

a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter generated by $(x_{\alpha})$ and $\mathcal{F_2}$ the filter generated by $(x_{h(\beta)})$. Then $\mathcal{F_2}$ is a refinement of $\mathcal{F_1}$.

b) Let $\mathcal{F}_1\subseteq \mathcal{F}_2$ two filters over $X$ and $(x_{\alpha})_{\alpha\in A}$ the net generated by $\mathcal{F}_1$, and $(y_{\beta})_{\beta\in B}$ the net generated by $\mathcal{F}_2$. Then $(y_{\beta})_{\beta\in B}$ is a subnet of $(x_{\alpha})_{\alpha\in A}$.

a) Let $F\in\mathcal{F}_1$. Then there exists $\alpha_0\in A$ with $\{x_{\alpha}:\alpha\ge\alpha_0\}\subseteq F$. By cofinality, there exists $\beta_0\in B$ such that $h(\beta_0)\ge\alpha_0$. By monotonicity, $\{x_{h(\beta)}:\beta\ge\beta_0\}\subseteq \{x_{\alpha}:\alpha\ge\alpha_0\}$ and so $F\in\mathcal{F}_2$. So $\mathcal{F_1}\subseteq \mathcal{F_2}$, hence $\mathcal{F_2}$ is a refinement of $\mathcal{F_1}$.

b) This is harder, so I think it is false. To clarify, here $A=\{(p,F):p\in F\in\mathcal{F}_1\}$, where $(p_1,F_1)\le (p_2,F_2)$ iff $F_2\subseteq F_1$ and $(x_{\alpha})_{\alpha\in A}$ is the net defined by $x_{(p,F)}=p$. Same way it is defined $(y_{\beta})_{\beta\in B}$. I don't really know how to show $(y_{\beta})_{\beta\in B}$ is a subnet of $(x_{\alpha})_{\alpha\in A}$.

Anyone knows if there's a counterexample?

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