On Borel $\sigma$-algebras of topologies generated by two-point selections

A two-point selection on a set $X$ is a function $f:[X]^2 \to X$ such that $f(F) \in F$ for every $F \in [X]^2$. It is known that every two-point selection $f:[X]^2 \to X$ induced a topology $\tau_f$ on $X$ by using the relation: $x \leq y$ if either $f(\{x,y\}) = x$ or $x = y$, for every $x, y \in X$. We are mainly concern with the two-point selections on the real line $\mathbb{R}$. In this paper, we study the $\sigma$-algebras of Borel, each one denoted by $\mathcal{B}_f(\mathbb{R})$, of the topologies $\tau_f$'s defined by a two-point selection $f$ on $\mathbb{R}$. We prove that the assumption $\mathfrak{c} = 2^{< \mathfrak{c}}$ implies the existence of a family $\{ f_\nu : \nu < 2^\mathfrak{c} \}$ of two-point selections on $\mathbb{R}$ such that $\mathcal{B}_{f_\mu}(\mathbb{R}) \neq \mathcal{B}_{f_\nu}(\mathbb{R})$ for distinct $\mu, \nu < 2^\mathfrak{c}$. By assuming that $\mathfrak{c} = 2^{< \mathfrak{c}}$ and $\mathfrak{c}$ is regular, we also show that there are $2^{2^\mathfrak{c}}$ many $\sigma$-algebras on $\mathbb{R}$ that contain $[\mathbb{R}]^{\leq \omega}$ and none of them is the $\sigma$-algebra of Borel of $\tau_f$ for any two-point selection $f: [\mathbb{R}]^2 \to \mathbb{R}$. Several examples are given to illustrate some properties of these Borel $\sigma$-algebras.