Dense and subspace dense subsets in finite-dimensional spaces

TL;DR

This paper investigates the density properties of subsets in finite-dimensional spaces, disproving a conjecture that dense sets necessarily intersect nontrivially dense subspaces, specifically in spaces of dimension two or higher.

Contribution

It provides a counterexample showing that not all dense subsets in finite-dimensional spaces contain a dense intersection with any nontrivial subspace.

Findings

Counterexample in or all nb2 showing the conjecture is false

Disproves the claim that dense sets always intersect densely with some subspace

Clarifies limitations of density properties in finite-dimensional spaces

Abstract

This note is motivated by the article of Bamerni, Kadets and Kili\c{c}man [J. Math. Anal. Appl. 435 (2), 1812--1815 (2016)]. We consider the remaining problem which claims that if $A$ is a dense subset of a finite dimensional space $X$, then there is a nontrivial subspace $M$ of $X$ such that $A\cap M$ is dense in $M$. We show that the above problem has a negative answer when $X=\mathbb{K}^{n}$ ($\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$) for every $n\geq 2$.