A Connection between Hyperreals and Topological Filters

Let $U$ be an absolute ultrafilter on the set of non-negative integers $\mathbb{N}$. For any sequence $x=(x_n{)}_{n\ge 0}$ of real numbers, let $U(x)$ denote the topological filter consisting of the open sets $W$ of $\mathbb{R}$ with $\{n\ge 0,x_n\in W\}\in U$. It turns out that for every $x\in {\mathbb{R}}^{\mathbb{N}}$, the hyperreal $\bar{x}$ associated to $x$ (modulo $U$) is completely characterized by $U(x)$. This is particularly surprising. We introduce the space $\tilde{\mathbb{R}}$ of saturated topological filters of $\mathbb{R}$ and then we prove that the set ${}^{\ast }\mathbb{R}$ of hyperreals modulo $U$ can be embedded in $\tilde{\mathbb{R}}$. It is also shown that $\tilde{\mathbb{R}}$ is quasi-compact and that ${}^{\ast }\mathbb{R}\setminus \mathbb{R}$ endowed with the induced topology by the space $\tilde{\mathbb{R}}$ is a separated topological space.