ON A TOPOLOGY DEFINED BY PRIMITIVE WORDS

Abstract

Let $f:X⟶X$ be a mapping. Consider $$\mathcal{P}\left(f\right)=\{O\subseteq X:f^{-1}(O)\subseteq O\}.$$ Then $\mathcal{P}\left(f\right)$ is an Alexandroff topology. A topological space $X$ is called a primal space if its topology coincides with a $\mathcal{P}\left(f\right)$ for some mapping $f:X⟶X$. Let $A$ be an alphabet and $A^{*}$ be the set of all finite words over $A$. A word is called primitive if it is not empty and not a proper power of another word. Let $u$ be a nonempty word; then there exists a unique primitive word $z$ and a unique integer $k\ge 1$ such that $u=z^k; z$ is called the primitive root of $u$; we denote by $z=p_A\left(u\right)$. It is convenient to set $p_A\left({\epsilon }_A\right)={\epsilon }_A$, where ${\epsilon }_A$ is the empty word over $A$. By a primitive primal space we mean a space $X$ that is homeomorphic to the subspace $A^{+}$ of $A^{*}$ equipped with the topology $\mathcal{P}\left(p_A\right)$ for some alphabet $A$. Our main result provides a structure theorem of primitive primal spaces.