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 $x=(x_n)_{n\geq 0}$ of real numbers, let $U(x)$ denote the topological filter consisting of the open sets $W$ of $\mathbb{R}$ with $\{n \geq 0, x_n \in W\} \in U$. It turns out that for every $x \in \mathbb{R}^{\mathbb{N}}$, the hyperreal $\overline{x}$ associated to $x$ (modulo $U$) is completely characterized by $U(x)$. This is particularly surprising. We introduce the space $\widetilde{\mathbb{R}}$ 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 $\widetilde{\mathbb{R}}$. It is also shown that $\widetilde{\mathbb{R}}$ is quasi-compact and that $^\ast\mathbb{R} \setminus \mathbb{R}$ endowed with the induced topology by the space $\widetilde{\mathbb{R}}$ is a separated topological space.