The Collaborative International Dictionary of English v.0.48:

Nearness \Near"ness\, n.
   The state or quality of being near; -- used in the various
   senses of the adjective.
   [1913 Webster]

WordNet (r) 3.0 (2006):

nearness
    n 1: the spatial property resulting from a relatively small
         distance; "the sudden closeness of the dock sent him into
         action" [syn: nearness, closeness] [ant: farawayness,
         farness, remoteness]

Moby Thesaurus II by Grady Ward, 1.0:

123 Moby Thesaurus words for "nearness":
   access, accession, accord, accordance, addition, adjunct, advance,
   advent, affairs, affiliation, affinity, afflux, affluxion,
   agreement, alikeness, alliance, analogy, aping, approach,
   approaching, appropinquation, approximation, appulse, assemblage,
   assimilation, association, avarice, bond, cheapness, chumminess,
   closefistedness, closeness, combination, coming, coming near,
   coming toward, community, comparability, comparison, confinement,
   conformity, connectedness, connection, contiguity, contrariety,
   copying, correspondence, crowdedness, dealings, deduction,
   disjunction, familiarity, filiation, flowing toward, forthcoming,
   hair, hairbreadth, hairsbreadth, hardfistedness, hoarding,
   homology, identity, illiberality, imitation, imminence,
   incapaciousness, incommodiousness, inseparableness, intercourse,
   intimacy, intimate acquaintance, junction, liaison, likeness,
   likening, limitation, link, linkage, linking, mateyness, metaphor,
   mimicking, miserliness, mutual attraction, narrow gauge,
   narrowness, nearing, oncoming, palliness, parallelism, parity,
   penny-pinching, propinquity, proximation, proximity, rapport,
   relatedness, relation, relations, relationship, resemblance,
   restrictedness, restriction, sameness, semblance, similarity,
   simile, similitude, simulation, slenderness, special affinity,
   stinginess, straitness, strictness, sympathy, tie, tie-in,
   tight purse strings, tight squeeze, tightfistedness, tightness,
   ungenerosity, union