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@PaulTaylor Thanks, I changed 'compactness' to 'separability' in title and gave an example in Real numbers. I'm struggling to frame the question without tying too closely to a single domain.
@Basil Thanks for the reference. Here are three "definability"-like properties: a model is definable if it can be constructed in some language; a model is "definably inhabited" if each of its elements is definable (e.g. the rationals or the algebraic reals, depending on language); and a model is "definably compact" if each of its elements is a sup of definable elements (e.g. the standard reals or the Bohm tree model). It is this last property I am interested in.
