A Knaster--Reichbach type theorem for graph structures

Abstract

We study the properties of a generic object $\mathbb{P}$ in the category of finite graphs. It turns out that this object, being topologically a Cantor set, has the Knaster--Reichbach type property. Namely, every homeomorphism and isomorphism $h\colon K\to L$ where $K$ and $L$ are nowhere dense closed sets in $\mathbb{P}$ and consisting only of isolated vertices in $K$ and $L$ can be extended to the autohomeomorphism and autoisomorphism of the whole graph $\mathbb{P}$.