Definitions of fineness:

• noun:   the quality of being very good indeed

  Example: "The inn is distinguished by the fineness of its cuisine"

• noun:   having a very fine texture

  Example: "The fineness of the sand on the beach"

• noun:   the property of being very narrow or thin

  Example: "He marvelled at the fineness of her hair"

• noun:   the quality of being beautiful and delicate in appearance

  Example: "The fineness of her features"