# About filters on real numbers

While trying to solve an open problem which I formulated earlier (currently "conjecture 1857" in this file), I met the following problem:

Let $\Delta_{\geq}+\alpha$ be the filter on $\mathbb{R}$ generated by the base $\mathopen[\alpha;\alpha+\epsilon\mathclose[$ where $\epsilon>0$.

Let $X$ be a set of negative numbers having zero as a limit point (that is $X$ has points arbitrarily near to zero).

Question Is $\bigcap_{\alpha\in X}(\Delta_{\geq}+\alpha)$ necessarily a subset of $\uparrow\mathopen]\alpha-\delta;\alpha\mathclose[$ for some $\delta>0$? ($\uparrow K$ denotes the principal filter generated by set $K$.)

• If you take $X=\lbrace-\frac{1}{n+1}\mid n\in\Bbb{N}\rbrace$, then for any sequence $\epsilon_n$ of positive real numbers the intersection you consider contains the set $$\bigcup_{n\in\Bbb{N}}\left[-\frac1{n+1},-\frac1{n+1}+\epsilon_n\right)$$ which can can have total length $\leq\sum_n\epsilon_n$ shorter than any $\delta>0$. So the intersection isn't included in any such principal filter. – Olivier Bégassat May 21 '17 at 21:57