1 Triplet of topological space modality
For the sake of uniformity, it is useful to reformulate our conclusions relating the topological space modality. Its three operators, that is, interior and closure together with negation, no (whose corresponding set is the empty set) form triplet of the quantifiers some, no, any, or, in spatial terms, somewhere, nowhere, and anywhere.
This spatial triplet has spatial sense which is distinct from the existential sense. “Somewhere,” “nowhere,” “anywhere” say nothing about existence, as alethic modal operators do. The spatial operators say about space: somewhere in space, nowhere in space, anywhere in space.
Sometime we can merge existential and spatial contexts, in the same manner as we can merge the axiological and deontic ones (e.g., to perform a duty is good, but duty and goodness belong, nevertheless, to different modalities).