An abstract characterization of menger algebras of strongly quasi-open multiplace maps

Abstract

Let X and Y be topological spaces. A map f : X → Y is quasi-open if Int(f (U)) ̸= ∅ for every non-empty open set U ⊂ X. We say that a map f : Xn → X is a strongly quasi-open map if for any non-empty set V ⊂ Xn for which all projections have non-empty interiors, the interior of f(V ) is non-empty. Let Q (Xn, X) denote the Menger algebra of strongly quasi-open maps from Xn to X with composition of functions: [fg1...gn] (a1, ..., an) = f (g1 (a1, ..., an), ..., gn (a1, ..., an)) A topological space X is said to be a TD-space if for every point ξ in X the set { ξ } r{ξ} is closed. Obviously, each TD-space is T0-space and each T1-space is TD-space. We call a topological space X a T + D -space if it is a TD-space with no one-point open sets and if for every point ξ in X and for every open set U containing ξ the set U ∩ ( X \ { ξ }) is not empty. Let X be T + D -space that has an open base, each element of which is an image of X under a quasi-open map and let Λ be a class of all such spaces. Theorem 1. Let X, Y ∈ Λ. The Menger algebras Q (Xn, X) and Q (Xn, X) are isomorphic if and only if the topological spaces X and Y are homeomorphic. Keywords: Menger algebra, strongly quasi-open map.