Infinite Lineability: On the Abundance of Dense Subspaces
Pedro Emerick, Luan Arjuna Belmonte
TL;DR
The paper investigates whether infinite (pointwise) dense-lineability is equivalent to (pointwise) dense-lineability, introducing $\alpha$-infinitely dense-lineability notions and analogous algebrability concepts. It proves the equivalence holds in all infinite-dimensional first-countable topological vector spaces and under certain weight constraints on the topology, but provides counterexamples to show the necessity of those conditions. Parallel results are developed for infinite dense-algebrability, including a criterion for strong dense-algebrability in complements $X\setminus Y$ of free subalgebras in free topological algebras. Together, these results clarify when large dense linear or algebraic structures exist inside pathological sets and supply constructions via transfinite methods.
Abstract
In this paper, we investigate the concept of infinite dense-lineability recently introduced by M. Calderón-Moreno, P. Gerlach-Mena and J. Prado-Bassas. We answer a question posed by the authors about the equivalence between infinite (pointwise) dense-lineability and (pointwise) dense-lineability. We prove that the equivalence always holds in first-countable topological vector spaces and under some assumptions about the weight of the topology. However, the equivalence is not always true, as shown in an example. Furthermore, we introduce the notions of infinite $(α,β)$-dense-lineability and infinite (strongly) dense-algebrability and obtain some analogous results in these cases. We also obtain a criterion for strongly dense-algebrability for sets of the form $X\setminus Y$, where $X$ is a free algebra and $Y$ is a free subalgebra of $X$.