论文标题
重新访问子空间和sublocales之间的关系
Revisiting the relation between subspaces and sublocales
论文作者
论文摘要
我们重新访问有关空间的子空间与其开放式集合所在地的子空间之间的连接的结果。我们提出的方法是基于这样的观察结果：对于每个语言环境$ l $它的空间sbrocales $ \ mathsf {sp} [\ mathsf {s}（l）] $形成一个coframe，对coframe $ \ mathsf {sob} [\ mathcal {pt p p} p p} $ \ Mathsf {pt}（l）$的子空间。我们表征了框架$ l $，以使$ \ mathsf {s}（l）$的空间sublocales完美地表示$ \ mathsf {pt}（l）$的子空间。我们证明了niefield和Rosenthal的无选择的弱版本和Rosenthal表征了这些框架，以使它们的所有sublocales都是空间的（即，$ \ Mathsf {pt}（l）$的清醒子空间完美地代表了$ l $的sublocales $）。我们通过使用基本素数的概念来做到这一点，该概念不依赖于每个元素上面的足够最小的素数的存在。我们将重新确认Simmons的结果，即$ω（x）$的sublocales完全代表其子空间正是散射的空间。我们将以强烈的素数重要形式来表征散落的空间。我们应用这些特征以表明，当$ l $是空间框架和coframe时，$ \ m athsf {pt}（l）$在且仅当它是$ t_d $时就会散布，并且仅当$ l $的所有prime of $ l $都是完全质量的。
We revisit results concerning the connection between subspaces of a space and sublocales of its locale of open sets. The approach we present is based on the observation that for every locale $L$ its spatial sublocales $\mathsf{sp}[\mathsf{S}(L)]$ form a coframe which is isomorphic to the coframe $\mathsf{sob}[\mathcal{P}(\mathsf{pt}(L))]$ of sober subspaces of $\mathsf{pt}(L)$. We characterize the frames $L$ such that the spatial sublocales of $\mathsf{S}(L)$ perfectly represent the subspaces of $\mathsf{pt}(L)$. We prove choice-free, weak versions of the results by Niefield and Rosenthal characterizing those frames such that all their sublocales are spatial (i.e., those such that the sober subspaces of $\mathsf{pt}(L)$ perfectly represent the sublocales of $L$). We do so by using a notion of essential prime which does not rely on the existence of enough minimal primes above every element. We will re-prove Simmons' result that spaces such that the sublocales of $Ω(X)$ perfectly represent their subspaces are exactly the scattered spaces. We will characterize scattered spaces in terms of a strong form of essentiality for primes. We apply these characterizations to show that, when $L$ is a spatial frame and a coframe, $\mathsf{pt}(L)$ is scattered if and only if it is $T_D$, and this holds if and only if all the primes of $L$ are completely prime.
