Closed copies of $\mathbb{N}$ in $\mathbb{R}^{\omega_1}$

TL;DR

This paper explores the existence and properties of closed copies of the natural numbers within powers of the real line, focusing on their embedding characteristics in the context of $C^*$- and $C$-embedding.

Contribution

It demonstrates that $ eal^{requency_1}$ contains closed copies of $ at$ that are not $C^*$-embedded, revealing new insights into embedding properties in infinite product spaces.

Findings

Existence of closed copies of $ at$ in $ eal^{requency_1}$

Identification of non-$C^*$-embedded copies of $ at$

Advances understanding of embedding properties in infinite-dimensional spaces

Abstract

We investigate closed copies of~$\mathbb{N}$ in powers of~$\mathbb{R}$ with respect to $C^*$- and $C$-embedding. We show that $\mathbb{R}^{\omega_1}$ contains closed copies of~$\mathbb{N}$ that are not $C^*$-embedded.