# Comparability

In mathematics, two elements *x* and *y* of a set *P* are said to be **comparable** with respect to a binary relation ≤ if at least one of *x* ≤ *y* or *y* ≤ *x* is true. They are called **incomparable** if they are not comparable.

A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable.

When classifying mathematical objects (e.g., topological spaces), two *criteria* are said to be comparable when the objects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they are comparable under the partial order ⊂. For example, the T_{1} and T_{2} criteria are comparable, while the T_{1} and sobriety criteria are not.