In this paper, we introduce and analyze the concept of T-equivalence spaces, which are defined as pairs \((X, E)\) , where \(X\) is a set and \(E\) is a T-equivalence relation on \(X\) . Our primary focus is on the structure of balls in these spaces and their topological properties. Specifically, we define balls as \(\alpha \) -cuts of a T-equivalence and prove that thus defined balls are open sets. In the metrizable case, we investigate how these relations can be expressed through a metric and an additive generator.