A Connection between Hyperreals and Topological Filters

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