In our thesis, we study the category of locales, a natural categorical approach to topology which generalises many topological constructions, e.g. sheaves. We show that sheaves on a locale also have an alternative perspective, since we can view them as a special case of sheaves on a site, i.e. a partially ordered set with a Grothendieck topology. Using this latter formulation we will give a striking application of the category of sheaves on a locale which refutes the continuum hypothesis if the latter is interpreted as a statement about that category. This argument contains the matematical core of Cohens proof of the compatibility of the negation of the continuum hypothesis with the axioms of Zermelo-Fraenkl set theory.
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