The quality or state of being contiguous; the property of sharing a border or existing adjacent to something else.
From 'contiguous' + '-ness' (noun suffix). The suffix converts the adjective into an abstract noun describing the quality itself, similar to how 'happiness' describes the quality of being happy.
This word is wonderfully awkward and almost never used—people just say 'contiguity'—which shows how English sometimes creates multiple words for the same concept and then decides one sounds better and keeps using that one!
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