Let $P$ be some topological property. A space $X$ is said to be $P$-situated in the bicompactification $bX$ if in $bX$, $X$ has a basis of open neighborhoods each of which possesses property $P$. In the class of normal and in the class of extremally disconnected spaces we give the intrinsic characteristics of spaces weakly paracompactly situated in their own Stone-Cech bicompactifications. We consider other applications of the concept of the $P$-situation of spaces in their bicompactifications.