A space is a pair ($X,$ $\Cal{K}$), where $X$ is a set and $K\subset exp X$. This paper gives some new characterizations for $S_i$-separation axioms in the space ($X$,$\Cal{K}$)($i=1,2$). As some corollaries of these results, some characterizations for $T_i$-separation axioms in the space ($X$,$\Cal{K}$) are obtained ($i=1,2$).