In a topological space, a point x is said to be a specialization of another point y if x is in the closure of y. Specialization closed subsets occur naturally when considering the notion of support in the Zariski topology in algebra/algebraic geometry. We will define them and show their classical use in classifying certain subcategories. This will allow us to give a characterization of Cohen-Macaulay local rings. Time permitting, we will also discuss some reductions of K-theoretic invariants (of derived categories with support).