双重否定何时是Scott连续的？
When is double negation Scott continuous?
AI总结：
针对T₀空间X的开集框架L，研究双重否定核的Scott连续性，证明其与X的离散性、有限性等价，推广结果到所有布尔核，给出X清醒且T₁的无点刻画。
AI中文摘要：
设L为T₀空间X的开集构成的框架，我们证明：若X是清醒（sober）且T₁的，则L上的双重否定核（nucleus）是Scott连续的当且仅当X是离散的。由此可得，若X还是紧的，则双重否定是Scott连续的当且仅当X是有限的。我们表明清醒性和T₁性这两个假设都是必要的，并将上述结果推广到L上所有布尔核（boolean nucleus）的情况。我们还给出了X是清醒且T₁的无点（pointfree）刻画，证明其等价于L上的Scott连续核是闭的。
英文摘要：
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.
