论文标题
borel和分析集
Borel and analytic sets in locales
论文作者
论文摘要
我们从毫无意义的拓扑背景下系统地从经典的描述集理论中系统地开发了基本概念的类似物。我们的起点是将框架$ \ MATHCAL {O}（X）$产生的免费完整布尔代数的元素开放为“ $ \ infty $ -borel sets”，以$ x $ $ x $。我们表明，该框架中可以将几个已知结果解释为经典描述性设定理论事实的直接类似物，包括例如Lusin分离，Lusin-Suslin和Baire类别定理。我们还证明了这些结果的几个扩展。我们对各种图像概念进行了详细的分析，并证明连续地图不必具有$ \ infty $ -borel图像。我们介绍了“分析$ \ inty $ -borel Locales”的类别作为定期完成的地方，该类别的图像和$ \ infty $ -Borel Maps的图像，并证明了几种有关分析集的经典结果的类似物，例如构成良好的分析关系的界限定理。我们还考虑了一个环境的“正$ \ infty $ -borel套件”，该语言环境是由不使用$ \ neg $的开放形成的。 实际上，我们使用$κ$ - $κ$κ$ -locales和$κ$ -Borel的套装，用于任意常规$ω_1\ lek \ lex \ le \ le \ infty $，从而将经典上下文作为特殊情况$κ=ω_1$。上述本地结果的基础是对出现$κ$ -Frames，$κ$ -Boolean代数等的各种已知和新方法的详细研究。特别是，我们介绍了一种新型的“双面posite”，用于呈现$（κ，κ，κ）$ - frames $ a $ a $和$ a $和$ a $ a $ a $ a^$ a^$ a^$ a^op，证明了$κ$ - ARY插值定理的$（κ，κ）$框架，该定理与上述分隔定理双重。
We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean algebra generated by the frame $\mathcal{O}(X)$ of opens to be the "$\infty$-Borel sets" in a locale $X$. We show that several known results in locale theory may be interpreted in this framework as direct analogs of classical descriptive set-theoretic facts, including e.g., the Lusin separation, Lusin-Suslin, and Baire category theorems for locales; we also prove several extensions of these results. We give a detailed analysis of various notions of image, and prove that a continuous map need not have an $\infty$-Borel image. We introduce the category of "analytic $\infty$-Borel locales" as the regular completion under images of the unary site of locales and $\infty$-Borel maps, and prove analogs of several classical results about analytic sets, such as a boundedness theorem for well-founded analytic relations. We also consider the "positive $\infty$-Borel sets" of a locale, formed from opens without using $\neg$. We in fact work throughout with $κ$-copresented $κ$-locales and $κ$-Borel sets for arbitrary regular $ω_1 \le κ\le \infty$, thereby incorporating the classical context as the special case $κ= ω_1$. The basis for the aforementioned localic results is a detailed study of various known and new methods for presenting $κ$-frames, $κ$-Boolean algebras, etc. In particular, we introduce a new type of "two-sided posite" for presenting $(κ, κ)$-frames $A$ (i.e., both $A$ and $A^{op}$ are $κ$-frames), and use this to prove a general $κ$-ary interpolation theorem for $(κ, κ)$-frames, which dualizes to the aforementioned separation theorems.
