Equivalence of localic symmetry to other separation axioms

Determine whether, for every locale L, the property of being symmetric—namely that S_c(L), the frame of joins of closed sublocales of L, is weakly subfit (equivalently, that every proper open sublocale of L is contained in a proper join of closed sublocales, or that every proper open dense sublocale of L is contained in a proper join of closed sublocales)—is equivalent to other established separation conditions for locales, in particular to T1-type properties studied in the literature on localic separation axioms.

Background

The paper introduces the notion of a symmetric locale, defined via equivalent conditions that involve the structure S_c(L) of joins of closed sublocales: S_c(L) being weakly subfit, or every proper open (or proper open dense) sublocale of L being contained in a proper join of closed sublocales.

Motivated by analyzing separation properties to detect failure of coframeness for S_c(L), the authors compare this notion to classical topological symmetry and T1 behavior, and pose whether the localic notion aligns with established localic separation axioms, especially T1-type properties previously studied.

References

Whether localic symmetry is equivalent to other known localic separation conditions (as studied in )—and in particular to T_1-type properties such as those examined in —remains an open question and will be explored in future work.

— Joins of closed sublocales are not always a coframe  (2510.00987 - Arrieta, 1 Oct 2025) in Subsection “What happens for spaces?” (Section 3), final paragraph