Axiom of Choice
The axiom of choice is equivalent to the statement that the product of a collection of non-empty sets is non-empty. The proof is easy enough: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.
The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice.
Read more about this topic: Product Topology
Famous quotes containing the words axiom of, axiom and/or choice:
“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“It is an axiom in political science that unless a people are educated and enlightened it is idle to expect the continuance of civil liberty or the capacity for self-government.”
—Texas Declaration of Independence (March 2, 1836)
“You didn’t have a choice about the parents you inherited, but you do have a choice about the kind of parent you will be.”
—Marian Wright Edelman (20th century)