In set theory, the Gelfand spectrum of a C*-algebra is only defined in the commutative case. Topos theory overcomes this restriction, especially through the oncept of a locale as the constructive analogue of a topological space. This approach unravels the logical structure of noncommutative spaces, which turns out to be intuitionistic.
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Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/111
[4] http://www.mpim-bonn.mpg.de/de/webfm_send/30/1
[5] http://www.mpim-bonn.mpg.de/de/webfm_send/30