Retract or Not: A Tale of Two Fans

TL;DR

This paper investigates when embeddings of Lelek and Cantor fans admit retractions, completing the classification of these cases and identifying conditions for retraction existence.

Contribution

It extends previous work by analyzing remaining fan combinations and characterizing embeddings that admit retractions.

Findings

Embeddings of Cantor fans into Lelek fans always admit retractions.

Embeddings of Lelek fans into Cantor fans do not admit retractions.

Existence of retractions depends on specific properties of the embeddings.

Abstract

Let $X$ be a Lelek fan or a Cantor fan and let $Y$ be a Lelek fan or a Cantor fan. In this paper, we study embeddings $ f: X \to Y $ that admit retractions from $ Y $ onto $ f(X)$. In 1989, W. J. Charatonik and J. J. Charatonik proved that if $X$ is a Lelek fan and $Y$ is a Cantor fan, then no embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. They also showed that if both $X$ and $Y$ are Cantor fans, then every embedding $f$ of $X$ into $Y$ admits such a retraction. In this paper, we address the two remaining cases. First, we consider the situation where $X$ is a Cantor fan and $Y$ is a Lelek fan. We prove that in this case, every embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. Second, we examine the case where both $X$ and $Y$ are Lelek fans. Here, we show that there exist embeddings $f$ that do admit a retraction from $Y$ onto $f(X)$, as well as embeddings that do not. For this latter case, we also identify additional properties of embeddings that ensure the existence of a retraction from $Y$ onto $f(X)$.