  In this paper we show that, in the definition of Alexandrov spaces with lower
or upper curvature bound, the original conditions can be replaced with much
weaker ones. For the purpose, we introduce `imaginary' comparison angles (and
`imaginary' angles), and the right or left bounded second derivative in the
support sense. As applications, we provide new proofs for the Doubling Theorem,
and the Globalization Theorem for complete or geodesic Alexandrov spaces with
lower curvature bound.
