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.
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Famous quotes containing the words sober and/or spaces:
“Learn to live well, or fairly make your will;
You’ve played, and loved, and eat, and drunk your fill:
Walk sober off; before a sprightlier age
Comes tittering on, and shoves you from the stage:
Leave such to trifle with more grace and ease,
Whom Folly pleases, and whose follies please.”
—Alexander Pope (1688–1744)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)