Duality of Sober Spaces and Spatial Locales
This section motivates and explains one of the most basic constructions of Stone duality: the duality between topological spaces which are sober and frames (i.e. complete Heyting algebras) which are spatial. This classical piece of mathematics requires a substantial amount of abstraction that usually tends to puzzle beginners. It should therefore be considered as graduate level mathematics. Some prior exposure to the basics of category theory is recommended, although a deep understanding of the concepts of adjunction and duality may well arise from examples such as the result below. Furthermore, concepts of topology and order theory are naturally involved as well, where the latter is probably more important for a thorough understanding.
Read more about this topic: Stone Duality
Famous quotes containing the words sober and/or spaces:
“Carlyle’s humor is vigorous and titanic, and has more sense in it than the sober philosophy of many another. It is not to be disposed of by laughter and smiles merely; it gets to be too serious for that: only they may laugh who are not hit by it.”
—Henry David Thoreau (1817–1862)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)