A point of a locale is a continuous map (in the sense of locales) to from the abstract point (seen as a locale whose corresponding frame is the frame of truth values).
If is the frame that corresponds to , then a point of is the same as a frame homomorphism from to the frame of truth values. This is the same as a completely prime filter in .
A point of is the same as a point (in the usual sense) of the topological space ; that is, the underlying set of is the set of points (in the sense above) of . (Thus, we call the space of points of .) Conversely, if is a topological space, then every point of determines a point of the locale of opens of . This map (which is a continuous map of topological spaces) is injective iff is (see separation axioms); it is a homeomorphism iff is sober.
Last revised on April 14, 2017 at 14:07:19. See the history of this page for a list of all contributions to it.