It is sometimes helpful to split a nonempty set up into disjoint smaller pieces. For example, we might have reason to split the integers into positive integers, negative integers, and the set containing zero alone. We often split the real numbers into rational numbers and irrational numbers, or we might want to break \(\mathbb {R}^{2}\) down into distinct vertical lines. All of these are examples of partitioning a space. In this chapter, we take a close look at partitions. The chapter concludes with a discussion of the relationship between a partition of a nonempty set X and the collection of the equivalence classes of a certain equivalence relation on X. In Tips on Putting it All Together we summarize some of the main strategies for working on mathematics.