中科院数学与系统科学研究院

数学研究所

学术报告

 集合论讨论班

报告人：申国桢（武汉大学）

时间：7月16日（星期二）14:00--15:00

地点：S813

题目：Boundedly finite-to-one functions

摘要：A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this talk, we prove in $\mathsf{ZF}$ (without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\mathcal{S}(A)$ to $\mathcal{S}_{\leq n}(A)$, where $\mathcal{S}(A)$ is the set of all pemutations of $A$ and $\mathcal{S}_{\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points; (2) For each infinite set $A$, there is no boundedly finite-to-one function from $\mathcal{B}(A)$ to $\mathrm{fin}(A)$, where $\mathcal{B}(A)$ is the set of all partitions of $A$ whose blocks are finite and $\mathrm{fin}(A)$ is the set of all finite subsets of $A$. This is joint work with Xiao Hu.