A locallic version of Hager’s metric-fine spaces is presented. A general definition of -fineness is given and various special cases are considered, notably all metric frames, complete metric frames. Their interactions with each other, quotients, separability, completion and other topological properties are discussed.
A bijective correspondence between strong inclusions and compactifications in the setting of -frames is presented. The category of uniform -frames is defined and a description of the Samuel compactification is given. It is shown that the Samuel compactification of a uniform frame is completely determined by the -frame consisting of its uniform cozero part, and consequently, any compactification of any frame is so determined.
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