Invertibility, Often

TL;DR

This paper demonstrates that in three mathematical categories, the set of isomorphisms is residual within the space of morphisms, highlighting their typicality in these contexts.

Contribution

It establishes that isomorphisms form a residual subset in various categories of continuous and measure-preserving maps, extending the understanding of their prevalence.

Findings

Isomorphisms are residual among surjective continuous maps on Cantor spaces.

Isomorphisms form a residual set among measure-preserving maps on Polish measure spaces.

Continuous, measure-preserving maps with good measures on Cantor spaces also have residual isomorphisms.

Abstract

By using a similar pattern of arguments, we show that in three categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at measure preserving maps on Polish measure spaces. Finally, we examine continuous, measure preserving maps on Cantor spaces equipped with so-called good measures.