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Some relations between cardinal invariants ofX andC(X) are established in the weak topology, whereC(X) is the space of continuous real-valued functions onX in the compact-open topology.

The following theorems are the main results of this paper. Theorem 1. Let : 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 be peripherally bicompact. Theorem 2. Let : be a closed mapping of a weakly paracompact p-space X. Then , where the set Y1 is σ-discrete in Y and the set is bicompact for each point γY0. An example is constructed of a weakly paracompact, locally compact, σ-paracompact space which is not normal and which cannot be mapped perfectly onto a space with a refining sequence of coverings. Bibliography: 22 items.