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For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...

Consider the following generalization of topological spaces: Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...

In [1, example C.1.2.8], a locale $Y$ (dense in another locale $X$) without any point is given. I fail to understand the point of such point-less locale - Why can't we identify those as the trivial ...

Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...