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:
“Far from the madding crowd’s ignoble strife,
Their sober wishes never learned to stray;
Along the cool sequestered vale of life
They kept the noiseless tenor of their way.”
—Thomas Gray (1716–1771)
“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
—Jean Baudrillard (b. 1929)