An embedding of the Cantor fan into the Lelek fan

TL;DR

This paper presents a new, straightforward method to embed the Cantor fan into the Lelek fan using Mahavier products, avoiding previous complex embeddings involving Erdős spaces.

Contribution

The authors introduce an alternative construction for embedding the Cantor fan into the Lelek fan using recent techniques of Mahavier products, simplifying prior approaches.

Findings

Successful embedding of the Cantor fan into the Lelek fan without Erdős space techniques.

Demonstrates the universality of the Lelek fan for smooth fans.

Provides a new method that simplifies the understanding of fan embeddings.

Abstract

The Lelek fan $L$ is usually constructed as a subcontinuum of the Cantor fan in such a way that the set of the end-points of $L$ is dense in $L$. It easily follows that the Lelek fan is embeddable into the Cantor fan. {It is also a well-known fact that the Cantor fan is embeddable into the Lelek fan, but this is less obvious. When proving this, one usually uses the well-known result by Dijkstra and van Mill that the Cantor set is embeddable into the complete Erd\"os space, and the well-known fact by Kawamura, Oversteegen, and Tymchatyn that the set of end-points of the Lelek fan is homeomorphic to the complete Erd\"os space. Then, the subcontinuum of the Lelek fan that is induced by the embedded Cantor set into the set of end-points of the Lelek fan, is a Cantor fan. In our paper, we give an alternative straightforward construction of a Cantor fan into the Lelek fan. We do not use the fact that the Cantor set is embeddable into the complete Erd\"os space and that it is homeomorphic to the set of end-points of the Lelek fan. Instead, we use our recent techniques of Mahavier products of closed relations to produce an embedding of the Cantor fan into the Lelek fan. Since the Cantor fan is universal for the family of all smooth fans, it follows that also the Lelek fan is universal for smooth fans.