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- W3091931641 abstract "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 $Omega (X)$ 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." @default.
- W3091931641 created "2020-10-15" @default.
- W3091931641 creator A5049309217 @default.
- W3091931641 date "2020-10-11" @default.
- W3091931641 modified "2023-10-01" @default.
- W3091931641 title "Revisiting the relation between subspaces and sublocales" @default.
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