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’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“On this narrow planet, we have only the choice between two unknown worlds. One of them tempts us—ah! what a dream, to live in that!—the other stifles us at the first breath.”
—Colette [Sidonie Gabrielle Colette] (1873–1954)