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

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?

1 Answer

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 $$E$$.