Lineability, spaceability, and additivity cardinals for Darboux-like functions

TL;DR

This paper introduces the maximal lineability cardinal number for subsets of topological vector spaces and explores its relation to other cardinal invariants, analyzing the lineability of various Darboux-like function families.

Contribution

It defines the maximal lineability cardinal number and relates it to existing invariants, applying these concepts to Darboux-like functions on Euclidean spaces.

Findings

Describes the relation between maximal lineability and other cardinal numbers.

Establishes lineability and spaceability of certain Darboux-like function families.

Provides a framework for analyzing lineability in topological vector spaces.

Abstract

We introduce the concept of {\em maximal lineability cardinal number}, $\mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $\HL(M)$, and lineability $\LL(M)$ of $M$. In particular, we will describe, in terms of $\LL$, the lineability and spaceability of the families of the following Darboux-like functions on $\real^n$, $n\ge 1$: extendable, Jones, and almost continuous functions.