Exponential Q-topological spaces

Abstract

In 2001, Escardo and Heckmann gave a characterization of exponential objects in the category TOP of topological spaces (without using categorical concepts), as those topological spaces (Y,T) for which there exists an splitting-conjoining topology on C((Y,T),S), where S is the Sierpinski topological space with two points 1 and 0 such that {1} is open but {0} is not. Motivated by Escardo and Heckmann, in this paper, we have obtained a characterization of exponential objects in the category Q-TOP of Q-topological spaces introduced by Solovyov in 2008 (where Q is a fixed member of a fixed variety of Ω-algebras), as those Q-topological spaces (Y,σ) for which there exists an splitting-conjoining Q-topology on [(Y,σ),(Q,〈idQ〉)], where (Q,〈idQ〉) is the Q-Sierpinski space. In the proofs, our approach is not category theoretic, only some basic concepts of Q-topological spaces are required. © 2019 Elsevier B.V.