# Topology of the space of bi-orderings of a free group on two generators

## TL;DR

This paper proves that the space of bi-orderings of a free group on two generators has no isolated points, extending known results about the space of left-orderings.

## Contribution

It establishes that the space of bi-orderings of a free group on two generators is topologically dense without isolated points, a new insight in group ordering topology.

## Key findings

- The space of bi-orderings of F_2 has no isolated points.
- The space of left-orderings of F_2 has no isolated points.
- Bi-orderings form a topologically dense subset in the space of orderings.

## Abstract

Let $G$ be a group. We can topologize the spaces of left-orderings $LO(G)$ and bi-orderings $O(G)$ of $G$ with the product topology. These spaces may or may not have isolated points. It is known that $LO(F_2)$ has no isolated points, where $F_2$ is a free group on two generators. In this paper we show that $O(F_2)$ has no isolated points as well.