Definitions of closeness:

• noun:   characterized by a lack of openness (especially about one's actions or purposes)
• noun:   a feeling of being intimate and belonging together

  Example: "Their closeness grew as the night wore on"

• noun:   the spatial property resulting from a relatively small distance

  Example: "The sudden closeness of the dock sent him into action"

• noun:   close or warm friendship
• noun:   extreme stinginess