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 of β⁢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 that β⁢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.

中文翻译：

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通过余数表征真实紧凑的语言环境

我们证明，当且仅当“余数”β⁢L∖L {\ beta L \ smallsetminus L}是错过的​​β⁢L{\ beta L}的零个子区域的连接时，一个完全规则的语言环境L是实紧凑的。这扩展了Mrówka的结果，该结果用Stone–Čech压实中的剩余空间来表征真正的紧凑空间。我们证明，当且仅当L是可数类型时，β⁢L∖L {\βL \ smallsetminus L}是Lindelöf，后者的可数区域设置与空间完全相同，可以用子区域替换子空间。