  This paper shows that, in the definition of Alexandrov space with lower
([BGP]) or upper ([AKP]) curvature bound, the original conditions can be
replaced with much weaker ones, which can be viewed as comparison versions of
the second variation formula in Riemannian geometry (and thus if we define
Alexandrov spaces using these weakened conditions, then the original definition
will become a local version of Toponogov's Comparison Theorem on such spaces).
As an application, we give a new proof for the Doubling Theorem by Perel'man.
