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: $x\le y$ if either $f(\{x,y\})=x$ or $x=y$, for every $x, y \in X$\mathrm{.}Wearemainlyconcernwiththetwo-pointselectionsontherealline$$\mathbb{R}$$\mathrm{.}Inthispaper,westudythe$\sigma$-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 ${ f_\nu : \nu < 2^\mathfrak{c} }$oftwo-pointselectionson$$\mathbb{R}$$suchthat$$\mathcal{B}${f\mu}($\mathbb{R}$) \neq $\mathcal{B}${f\nu}($\mathbb{R}$)$ for distinct $\mu ,\nu <2^{\mathfrak{c}}$. By assuming that $\mathfrak{c} = 2^{< \mathfrak{c}}$and$\mathfrak{c}$isregular,wealsoshowthatthereare$2^{2^\mathfrak{c}}$many$\sigma$-algebrason$$\mathbb{R}$$thatcontain$[$\mathbb{R}$]^{\leq \omega}$andnoneofthemisthe$\sigma$-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.