Remote sublocales

Abstract

We use the notion of a remote collection of a Tychonoff space to define a remote sublocale of any locale. Our definition is conservative, in the sense that, a subset is remote if and only if the sublocale it induces is remote in the locale of opens. We characterize remote sublocales and show that the Booleanization of a locale is the largest remote sublocale of the locale, a result with no pointset topological counterpart. We study localic maps that send remote sublocales back and forth. It turns out that the localic maps which preserve remote sublocales are precisely those with skeletal left adjoints.

Mathematics Subject Classification (2020):

• 06D22

Key words:

• Remote sublocale, remote point
• Booleanization
• nowhere dense sublocale
• frame homomorphism
• skeletal map
• localic map