We find universal functions for the class of lower semi-continuous (LSC) functions with at most n-dimensional domain. In an earlier paper we proved that a space is almost n-dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most n-dimensional domain. We conclude that the class of almost n-dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdös space. © 2007 American Mathematical Society Reverts to public domain 28 years from publication.