Wajsberg algebras are the algebraic counterpart of Łukasiewicz logic that is defined with axioms which characterize implication and negation and with Modus Ponens as a rule [\textit{A. J. Rodriguez}, Un estudio algebraico de los cálculos proposicionales de Łukasiewicz. Ph. D. Thesis, Univ. Barcelona (1980)]. In this paper, Wajsberg algebras are considered as Kleene algebras and bounded distributive lattices. The aim of this paper is to develop a duality theory for Wajsberg algebras, extending the duality between Kleene algebras and some ordered topological spaces that is considered by \textit{W. Cornish} and \textit{P. Fowler} [J. Austral. Math. Soc., Ser. A 27, 209-220 (1978; Zbl 0403.06010)]. A Wajsberg space is characterized as Kleene space over a family of partial functions satisfying a number of algebraic and topological properties.