Speaker
Description
In this talk we show that a topological pair of a chainable graph and the set of its endpoints has computable type.
The notion of a chainable graph is inspired by the notion of a graph - a set which consists of finitely many arcs such that distinct arcs intersect in at most one endpoint. It is known that if $G$ is a graph and $E$ set of all endpoints of $G$, that then $(G,E)$ has computable type.
We define the following: Suppose $A$ is a topological space. Let $V \subseteq A$ be a finite subset of $A$ and let $\mathcal{K}$ be a finite family of pairs $(K, \{a,b\})$ where $a,b \in V$, $a \neq b$ and $K \subseteq A$ is a continuum chainable from $a$ to $b$.
Suppose
$$A = V \cup \bigcup_{(K,\{a,b\}) \in \mathcal{K}}K$$
and that the following holds:
if $(K, {a,b})$, $(L,{c,d}) \in \mathcal{K}$ and $K \neq L$, then $\operatorname{card} (K \cap L) < \aleph_0$.
Then the triple $(A, \mathcal{K}, V)$ is called a **chainable graph.** We say that $a \in V $ is an **endpoint** of $(A, \mathcal{K}, V)$ if there exist only one $K \subseteq A$ and at least one $b \in V$ such that $(K, {a,b}) \in \mathcal{K}$.
For such an object, it holds:
If $(A,\mathcal{K},V)$ is a chainable graph and $B$ is the set of all its endpoints, then $(A,B)$ has computable type.
