curiousness

1. Carefulness; painstaking. [Obs.]

My father's care
With curiousness and cost did train me up.

2. The state of being curious; exactness of workmanship; ingenuity of contrivance.

3. Inquisitiveness; curiosity.

noun

1. Mental acquisitiveness: curiosity, inquisitiveness, interest. Idioms: thirst for knowledge. See investigate.
2. Undue interest in the affairs of others: curiosity, inquisitiveness. Informal nosiness, snoopiness. See investigate.

The noun has 2 meanings:

Meaning #1: a state of active curiosity
  Synonym: inquisitiveness

Meaning #2: the quality of being alien or not native
  Synonyms: foreignness, strangeness