In many tensor triangulated categories the thick subcategories of the compact objects and the localizing subcategories of the whole category are classified by topological spaces with the same underlying set. Examples include the derived category of a commutative noetherian ring and the stable module category of a finite group. A well-generated triangulated category is generated by a set of \alpha-compact objects for some regular cardinal \alpha. Under some mild conditions the \alpha-localizing subcategories of \alpha- compact objects are also classified by a topological space. In my talk I explain the connection between the topological spaces classifying the thick subcategories of the compact objects, the \alpha-localizing subcategories of the \alpha-compact objects, and the localizing subcategories. This is joint work with Henning Krause.