A Connection between Hyperreals and Topological Filters

TL;DR

This paper explores the deep connection between hyperreal numbers and topological filters, introducing a new space of saturated filters and demonstrating how hyperreals can be embedded within it, revealing surprising characterizations.

Contribution

It introduces the space of saturated topological filters of  and shows that hyperreals can be embedded into this space, establishing a novel link between nonstandard analysis and topology.

Findings

Hyperreals are characterized by associated topological filters.

The space of saturated filters  is quasi-compact.

Hyperreals form a separated space within .

Abstract

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.