ON A TOPOLOGY DEFINED BY PRIMITIVE WORDS

Abstract

Let f:X⟶X be a mapping. Consider P(f)={O⊆X:f-1(O)⊆O}. Then P(f) is an Alexandroff topology. A topological space X is called a primal space if its topology coincides with a P(f) 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≥1 such that u=zk; z is called the primitive root of u; we denote by z=pA(u). It is convenient to set pA(εA)=εA, where εA 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 P(pA) for some alphabet A. Our main result provides a structure theorem of primitive primal spaces.