A T_2 space, also known as a Hausdorff space, is a type of topological space in which any two distinct points can be separated by neighborhoods. This means that for any two points, there exist open sets containing each point that do not overlap. This property is important in topology because it ensures that points can be distinguished from one another. In a T_2 space, the separation of points leads to useful properties in analysis and geometry. For example, limits of sequences are unique in T_2 spaces, which is crucial for defining continuity and convergence. This makes T_2 spaces a foundational concept in modern mathematics.