The gleason cover of a topos, II

In [13], it was observed that the topos Shv(X) of sheaves on a topological space X satisfies De Morgan’s law iff X is extremally disconnected, and it was claimed that most of the occurrences of extremally disconnected spaces in general topology and functional analysis could in fact be explained as appeals to De Morgan’s law in the internal logic of Shv(X). In this and a subsequent paper [17], we set out to justify this claim in the case of a particular topological construction involving extremally disconnected spaces: the Gleason cover. In [7], A.IM. Gleason proved the following two results:

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