On the Cauchy integral theorem and Polish spaces

Abstract

We prove that if a continuous function $f$ in an open subset $U\subset\mathbb{C}$ is analytic in $U\setminus X$, where $X\subset U$ is a Polish space having characteristic system $(k,n)\in\mathbb N_0\times\mathbb N$, then the complex line integral of $f$ along the boundary of any triangle in $U$ vanishes.