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 WordNet
add/edit