Stone Spaces
Each Boolean algebra B has an associated topological space, denoted here S(B), called its Stone space. The points in S(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra. The topology on S(B) is generated by a basis consisting of all sets of the form
where b is an element of B.
For any Boolean algebra B, S(B) is a compact totally disconnected Hausdorff space; such spaces are called Stone spaces (also profinite spaces). Conversely, given any topological space X, the collection of subsets of X that are clopen (both closed and open) is a Boolean algebra.
Read more about this topic: Stone's Representation Theorem For Boolean Algebras
Famous quotes containing the words stone and/or spaces:
“I passed a tomb among green shades
Where seven anemones with down-dropped heads
Wept tears of dew upon the stone beneath.”
—Unknown. The Thousand and One Nights.
AWP. Anthology of World Poetry, An. Mark Van Doren, ed. (Rev. and enl. Ed., 1936)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)