ON A TOPOLOGY DEFINED BY PRIMITIVE WORDS

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 = z^k; z is called the primitive root of u; we denote by z = p_A (u). It is convenient to set p_A (ε_A) = ε_A, where ε_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 P(p_A) for some alphabet A. Our main result provides a structure theorem of primitive primal spaces.