Exponential Q-topological spaces

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 Omega-algebras), as those Q-topological spaces (Y, sigma) for which there exists an splitting-conjoining Q-topology on [(Y, sigma), (Q, < id(Q)>)], where (Q, < id(Q)>) is the Q-Sierpinski space. In the proofs, our approach is not category theoretic, only some basic concepts of Q-topological spaces are required. (C) 2019 Elsevier B.V. All rights reserved.

Automated summary

The article presents the characterization of exponential objects in topological spaces by Escardo and Heckmann, and further extends it to Q-topological spaces, emphasizing that the proof method is not based on category theory.