Fans, phans and pans

TL;DR

The paper addresses when a 1-dimensional continuum $X$ that is a union of arcs meeting only at a single point $v$ must be a fan; it proves the converse of a classical result under two extra assumptions, namely that $X$ is 1-dimensional and hereditarily unicoherent, so $X=\bigcup_{L\in \mathcal{L}}L$ with $L_1\cap L_2=\{v\}$ implies $v$ is the unique ramification point. It develops a comprehensive pan/phan framework around ramification pairs $(v,\mathcal{L})$, using limit properties like $\limsup [x_n,y_n]_{(v,\mathcal{L})}$ and order-preserving conditions to relate arc unions to fan structure and classify them as Carolyn pans, Matea phans, or Jure phans. The main contributions include a proof of the converse under the stated hypotheses, a detailed taxonomy of pan/phan notions and their interactions (including the existence of non-smooth Jure phans), and a set of open problems about equivalences among phans and related structures. Collectively, these results clarify when simple arc-union representations guarantee full fan structure and advance understanding of continua and their hyperspaces.

Abstract

A fan is an arcwise-connected continuum, which is hereditarily unicoherent and has exactly one ramification point. Many of the known examples of fans were constructed as 1-dimensional continua that are unions of arcs which intersect in exactly one point. Borsuk proved in 1954 that each fan is a 1-dimensional continuum which is the union of arcs intersecting in exactly one point. But it is not yet known if this property is equivalent to being a fan. In this paper, we show that under two additional assumptions, every such union of arcs is a fan.

Figures (2)

• Figure 1: $X=\bigcup_{x\in A}L_{x}$
• Figure 2: The non-smooth fan $X$ from Example \ref{['maj']}

Theorems & Definitions (45)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 2.7
• Example 3.1
• Lemma 3.2
• proof