An ultranet $x_\lambda$ is frequentely in $Y$ if and only if it is residually too.

Question

Definition

If $x_\lambda$ is a net from a directed set $\Lambda$ into $X$ and if $Y$ is a subset of $X$ then we say that $x_\lambda$ is redisually in $Y$ if there exsit $\lambda_0\in\Lambda$ such that $X_\lambda\in Y$ for any $\lambda\ge\lambda_0$

Definition

If $x_\lambda$ is a net from a directed set $\Lambda$ into $X$ and if $Y$ is a subset of $X$ then we say that $x_\lambda$ is frequently in $Y$ if for any $\lambda\in\Lambda$ there exist $\lambda_0\ge\lambda$ such that $x_{\lambda_0}\in y$

What shown belove is a reference from "General Topology" by Stephen Willard

So I want to discuss the claim for which if an ultranet is frequently in $E$ then it is residually in $X-E$.

Cleraly if $x_\lambda$ is a net residualliy in $Y$ then for any $\overline{\lambda}\in\Lambda$ there exist $\lambda_0$ such that if $\lambda\ge\lambda_0$ then $x_\lambda\in Y$ and so if we pick $\overline{\lambda}_0\in\Lambda$ such that $\overline{\lambda},\lambda_0\le\overline{\lambda}_0$ (we can do this since $\Lambda$ is a directed set) then it follows that $x_{\overline{\lambda}_0}\in Y$ and $\overline{\lambda}_0\ge\overline{\lambda}$ so that $x_\lambda$ is frequentely in $Y$.

So clearly any ultranet is a net and so for what we have proved above if an ultranet is residually in $E$ then it is frequentely too.

However I can't prove the inverse implication so I ask to do it. Could someone help me, pease?

Answer 1

Suppose that $\nu=\langle x_d:d\in D\rangle$ is an ultranet that is frequently in some set $E$. Then for each $d_0\in D$ there is a $d\in D$ such that $d_0\le d$ and $x_d\in E$, so $\nu$ cannot be residually in $X\setminus E$. But $\nu$ is an ultranet, so by definition it is either residually in $E$ or in $X\setminus E$, and since it is not residually in $X\setminus E$, it must be residually in $E$.