No abstract

No abstract

Abstract. 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 Erdős space.Unless stated otherwise all topologies in this note are assumed to be separable and metrizable.Let n be a nonnegative integer and let X be a topological space. We say that X is almost n-dimensional if there exists a topology W on X such that dim(X, W) ≤ n, W is weaker than the given topology on X, and every point of X has a neighbourhood basis in X consisting of sets that are closed in (X, W). If we substitute n = 0, then we get the notion of an almost zero-dimensional space, which concept has been introduced by Oversteegen and Tymchatyn [16]; see also [13] Both spaces were introduced and shown to be one-dimensional by Paul Erdős