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We prove that a completely regular locale  L  is realcompact if and only if the “remainder”β  ⁢  L   ∖  L{\beta L\smallsetminus L}    is the join of the zero-sublocales ofβ  ⁢  L{\beta L}    that miss  L . This extends a result of Mrówka which characterizes realcompact spaces in terms of their remainders in Stone–Čech compactifications. We prove thatβ  ⁢  L   ∖  L{\beta L\smallsetminus L}    is Lindelöf if and only if  L  is of countable type, where the latter is defined for locales exactly as for spaces, subject to replacing subspaces with sublocales.

Characterizing realcompact locales via remainders