论文标题
在Borel $σ$ - 两点选择产生的拓扑的代数
On Borel $σ$-algebras of topologies generated by two-point selections
论文作者
论文摘要
在集合$ x $上进行两点选择是函数$ f：[x]^2 \ to x $，因此[x]^2 $中的每个$ f \ in f $ in f $ in f $ in f $ in f $ f（f）\。众所周知，每两个两点选择$ f：[x]^2 \ to x $ to x $通过使用该关系在$ x $上诱导了$τ_f$，如果$ f（\ {x，y \}）= x $ f（\ {x，y \}）= x $或$ x = y $，则每$ x $ x $ x $ x $ x $ x，y in x $ in x $ in x $。我们主要关注实际行$ \ mathbb {r} $上的两点选择。在本文中，我们研究了Borel的$σ$ - 代数，每一个用$ \ Mathcal {b} _f（\ Mathbb {r}）$表示，拓扑为$τ_f$的$ \ \ Mathbb {r} $的两点选择$ f $。我们证明了假设$ \ mathfrak {c} = 2^{<\ mathfrak {c}} $意味着一个家庭$ \ {f_ν：ν<2^\ mathfrak {c} {c} \} $ oon $ \ \ m athbb {r} $的两点选择$ \ MATHCAL {B} _ {F_μ}（\ Mathbb {r}）\ neq \ neq \ Mathcal {b} _ {f_ν}（\ Mathbb {r}）$ for Divstance $μ，$μ，n <2^\ Mathfrak {c} $。通过假设$ \ Mathfrak {C} = 2^{<\ Mathfrak {C}} $和$ \ Mathfrak {C} $是常规的，我们还表明，$ 2^{2^{2^\ Mathfrak {c}} $许多$σ$ -Algebras on $ \ mathbb} $ [\ MATHBB {r}]^{\ leqω} $，它们都不是$σ$ -SALGEBRA borel的$τ_f$ of $τ_f$用于任何两点选择$ f：[\ mathbb {r}]^2 \ to \ mathbb {r} $。给出了几个示例来说明这些Borel $σ$ - 代数的某些属性。
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 $τ_f$ 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 $σ$-algebras of Borel, each one denoted by $\mathcal{B}_f(\mathbb{R})$, of the topologies $τ_f$'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 $\{ f_ν: ν< 2^\mathfrak{c} \}$ of two-point selections on $\mathbb{R}$ such that $\mathcal{B}_{f_μ}(\mathbb{R}) \neq \mathcal{B}_{f_ν}(\mathbb{R})$ for distinct $μ, ν< 2^\mathfrak{c}$. By assuming that $\mathfrak{c} = 2^{< \mathfrak{c}}$ and $\mathfrak{c}$ is regular, we also show that there are $2^{2^\mathfrak{c}}$ many $σ$-algebras on $\mathbb{R}$ that contain $[\mathbb{R}]^{\leq ω}$ and none of them is the $σ$-algebra of Borel of $τ_f$ for any two-point selection $f: [\mathbb{R}]^2 \to \mathbb{R}$. Several examples are given to illustrate some properties of these Borel $σ$-algebras.
