  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.
