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$\sigma $-frame; nearness; Samuel compactification
We introduce the structure of a nearness on a $\sigma $-frame and construct the coreflection of the category ${\mathbf N\sigma Frm}$ of nearness $\sigma $-frames to the category ${\mathbf KReg\sigma Frm}$ of compact regular $\sigma $-frames. This description of the Samuel compactification of a nearness $\sigma $-frame is in analogy to the construction by Baboolal and Ori for nearness frames in [1] and that of Walters for uniform $\sigma $-frames in [11]. We also construct the uniform coreflection of a nearness $\sigma $-frame, that is, the coreflection of the category of ${\mathbf N\sigma Frm}$ to the category U$\sigma $Frm of uniform $\sigma $-frames.
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