Definition: Connected and Disconnected

An open set S is called disconnected if there are two open, non-empty sets U and V such that:
  1. U V = 0
  2. U V = S

A set S (not necessarily open) is called disconnected if there are two open sets U and V such that

  1. (U S) # 0 and (V S) # 0
  2. (U S) (V S) = 0
  3. (U S) (V S) = S

If S is not disconnected it is called connected.

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