The following theorems are the main results of this paper. Theorem 1.Let$f\colon X\to Y$be a closed mapping of the weakly paracompact$p$-space$X$. In order that the space$Y$be weakly paracompact and plumed, it is necessary and sufficient that the mapping$f$be peripherally bicompact.\smallskipTheorem 2.{\it Let$f\colon X\to Y$be a closed mapping of a weakly paracompact$p$-space$X$. Then$Y=Y_0\cup Y_1,$where the set$Y_1$is$\sigma$-discrete in$Y$and the set$f^{-1}y$is bicompact for each point$y\in Y_0$.}An example is constructed of a weakly paracompact, locally compact,$\sigma$-paracompact space which is not normal and which cannot be mapped perfectly onto a space with a refining sequence of coverings.Bibliography: 22 titles.