Compactifications of unstable N\"obeling spaces

TL;DR

This paper constructs an embedding of a N"obeling space into a Menger space of the same codimension, solving a long-standing open problem in topological embedding theory.

Contribution

It provides the first known embedding of a N"obeling space into a Menger space in codimension 2, addressing an open problem from 1978.

Findings

Successfully embeds N"obeling space into Menger space in codimension 2

Resolves a 45-year-old open problem in topology

Advances understanding of space embeddings in geometric topology

Abstract

We construct and embedding of a N\"obeling space $N^n_{n-2}$ of codimension $2$ into a Menger space $M^n_{n-2}$ of codimension $2$. This solves an open problem stated by R.~Engelking in 1978 in codimension~$2$.