Large free linear algebras of real and complex functions

Abstract

Let $X$ be a set of cardinality $κ$ such that $κ^ω=κ$. We prove that the linear algebra $\mathbb{R}^X$ (or $\mathbb{C}^X$) contains a free linear algebra with $2^κ$ generators. Using this, we prove several algebrability results for spaces $\mathbb{C}^\mathbb{C}$ and $\mathbb{R}^\mathbb{R}$. In particular, we show that the set of all perfectly everywhere surjective functions $f:\mathbb{C}\to\mathbb{C}$ is strongly $2^\mathfrak{c}$-algebrable. We also show that the set of all functions $f:\mathbb{R}\to\mathbb{R}$ whose sets of continuity points equals some fixed $G_δ$ set $G$ is strongly $2^\mathfrak{c}$-algebrable if and only if $\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself.