The three most general categories in which the existence of an envelope of holomorphy is guaranteed are the category of unbranched Riemann domains over \({\mathbb{C}}^ n\), the category of branched Riemann domains over \({\mathbb{C}}^ n\), and the category of holomorphically convex complex spaces. - Branched Riemann domains over \({\mathbb{C}}^ n\) are holomorphically spreadable. There are many important non-spreadable spaces, however, which also have an envelope of holomorphy. The simplest example is obtained by blowing up the origin in \({\mathbb{C}}^ 2\); another such space is a counterexample of Skoda to the Serre problem. The purpose of this paper is to show that every connected normal complex space X whose separation relation \(R^ X\) is locally semiproper has an envelope of holomorphy. The equivalence relation \(R^ X\) is given by identifying those points of X which cannot be separated by global holomorphic functions. Since \(R^ X\) is locally proper for a holomorphically spreadable space X, the category of normal spaces considered in this paper contains the subcategory of branched Riemann domains over \({\mathbb{C}}^ n\). Two interesting properties of the envelope of holomorphy H(X) of connected normal spaces X with a locally semiproper separation relation are that dim H(X)\(\leq \dim X\) holds and that H(X) is holomorphically spreadable, even though X need not be spreadable. Examples for which dim H(X)\(<\dim X\) is true are also mentioned.