Zermelo-Fraenkel set theory with Choice (ZFC) is a foundational system for mathematics that describes how sets can be constructed and manipulated. It consists of a collection of axioms that define the properties and relationships of sets, such as the existence of empty sets, the ability to form unions, and the concept of subsets. The Axiom of Choice is a crucial component of ZFC, asserting that for any collection of non-empty sets, it is possible to select one element from each set. This axiom has significant implications in various areas of mathematics, including analysis and topology, and is essential for proving many important theorems.