Contrasting Symmetric Porosity and Porosity

Abstract

It is known that the following two fundamental properties of porosity fail for symmetric porosity: 1) Every nowhere dense set A contains a residual subset of points x at which A has porosity 1. 2) If A is a porous set and 0 < p < 1, then A can be written as a countable union of sets, each of which has porosity at least p at each of its points . Here we explore the somewhat surprising extent to which these properties fail to carry over to the symmetric setting and investigate what symmetric analogs do hold.