On the Cauchy Integral Theorem and Polish spaces

Abstract

We prove that if a function $f$ is continuous in an open subset $U\subset \mathbb{C}$ and analytic in $U\setminus X$, where $X\subset U$ is a Polish space having characteristic system $(i,n)$, such that $i\in \{0,1\}$ and $n\in \mathbb{N}$, then the complex integral line of $f$ along the boundary of any triangle in $U$ vanishes.