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arXiv · 2608.17107

When is double negation Scott continuous?

Guram Bezhanishvili·Sebastian D. Melzer
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Abstract

Let $L$ be the frame of opens of a $T_0$-space $X$. We prove that if $X$ is sober and $T_1$, then the double negation nucleus on $L$ is Scott continuous iff $X$ is discrete. It follows that if, in addition, $X$ is compact then double negation is Scott continuous iff $X$ is finite. We show that both the sober and $T_1$ assumptions are essential, and generalize the above results to all boolean nuclei on $L$. A pointfree characterization of when $X$ is sober and $T_1$ is also given by proving that it is equivalent to Scott continuous nuclei on $L$ being closed.

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Guram Bezhanishvili, Sebastian D. Melzer. 2026-08-17. When is double negation Scott continuous?. https://arxiv.org/abs/2608.17107

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