Abstract
We introduce and investigate a separation property that we call an R1-Urysohn property. A topological space (X, τ ) is called R1-Urysohn if, whenever x, y ∈ X satisfy cl x ̸= cl y, there exist disjoint open sets U, V ∈ τ such that cl x ⊆ clU and cl y ⊆ cl V. The definition is formulated in terms of closures of singleton sets, and therefore naturally interacts with the classical R0 and R1 separation axioms. We establish basic relationships with R0, R1, Hausdorff, regular, and Urysohn spaces. We prove that every Urysohn space is R1-Urysohn and that every R1 space is R1-Urysohn. Several examples are given to show that the new condition does not automatically imply the classical R1 or Hausdorff properties. We also investigate the behavior of the property under subspaces, products, continuous maps, and quotient constructions.