Large free linear algebras of real and complex functions

Abstract

Let $X$ be a set of cardinality $\kappa$ such that ${\kappa }^{\omega }=\kappa$. We prove that the linear algebra ${\mathbb{R}}^X$ (or ${\mathbb{C}}^X$) contains a free linear algebra with $2^{\kappa }$ 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 $G_{\delta }$ set $G$ is strongly $2^{\mathfrak{c}}$-algebrable if and only if $\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself.

References in corpus (1)

• An undecidable case of lineability in R^R