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Synonyms for Closeness (same or very similar meaning)

WordNet sense 1 (a feeling of being intimate and belonging together):
intimacy

WordNet sense 2 (the quality of being close and poorly ventilated):
stuffiness

WordNet sense 3 (the spatial property resulting from a relatively small distance):
nearness

WordNet sense 4 (extreme stinginess):
meanness, niggardliness, parsimony, tightness, minginess, niggardness, parsimoniousness, tightfistedness

WordNet sense 5 (characterized by a lack of openness (especially about one's actions or purposes)):
secretiveness

WordNet sense 6 (close or warm friendship):
familiarity


From the ODE community, based on WordNetadd/edit