This paper provides a theory of infinite dimensional holomorphy which allows nonconvex domains U with empty interior. For an example of such U, consider the subset of H(\({\mathbb{C}},{\mathbb{C}})\) (usual Fréchet space) formed by all never vanishing maps \(\phi: {\mathbb{C}}\to {\mathbb{C}}\). Then f: \(U\to U\), \(f(\phi)=1/\phi\), is an example of a function excluded from previous complex differentiation theories while now emerging as holomorphic map. The new calculus is obtained by verifying that the category \({\mathcal C}_ h\) of holological spaces [\textit{A. Kriegl} and \textit{L. D. Nel}, Cah. Topologie Géom. Différ. Catégoriques 26, 273-309 (1985; Zbl 0581.46041)] upholds the axioms postulated in [\textit{L. D. Nel}, Infinite dimensional calculus allowing nonconvex domains with empty interior, Monatsh. Math. 110, 145-166 (1990)].