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The paper "Grothendieck Topologies on Posets" by A.J. Lindenhovius shows thatwhen $\mathbf{P}$ is an Artinian poset and $\mathbf{E}$ is the topos$\mathbf{Set}^\mathbf{P}$ then there are bijections between the set of subsetsof $\mathbf{P}$, the set of Grothendieck topologies on $\mathbf{E}$, and theset of nuclei on the Heyting Algebra $\mathrm{Sub}(1_\mathbf{E})$. It alsoshows that there are nice formulas for converting between subsets, Grothendiecktopologies, and nuclei, but the formula for converting a nucleus to a subset isnot spelled out explicitly. These notes fix that gap.

On a formula that is not in "Grothendieck Topologies in Posets"