Porosity of the Extendable Connectivity Function Space

Abstract

Let $I = [0,1]$, and let $Ext(I)$ or $Ext$ denote the subspace of all extendable connectivity functions $f:I \to {\mathbb R}$ with the metric of uniform convergence on $I^{\mathbb R}$. We show that $Ext$ is porous in the almost continuous function space $AC$ by showing that the space $AC \cap PR$ of all almost continuous functions with perfect roads is porous in $AC$. We also show that for $n >1$, the subspace $Ext({\mathbb R}^n)$ of all extendable connectivity functions $f:{\mathbb R}^n \to {\mathbb R}$ is a boundary set in the Darboux function space $D({\mathbb R}^n)$.