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Abstract
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This investigation is devoted to three types of topological spaces, paracompact spaces, collectionwise normal spaces and collectionwise Hausdorff spaces. A $T_1$ space $X$ is collectionwise normal if every discrete collection $\mathcal F$ of closed subsets of $X$ has an open separation. Similarly topological space $X$ is collectionwise Hausdorff if every closed discrete set $F$ of $X$ has an open separation.
At First, in 1944, J. Dieudonné proved that a topological space $X$ is paracompact if it is Hausdorff and every open cover $\mathcal U$ of $X$ has a locally finite, open refinement $\mathcal V$. Paracompactness has been grown popularity in world of mathematics because of its widest applications in many different branches of mathematics. In chapters III and V we review some of equivalent definitions of paracampactness. Also we study some conditions to extend this notion in its hereditary form and the sum and the cartesian product of paracompact spaces. Also we investigates the conditions in which a mapping takes a paracompact (collectionwise normal, metacompact,...) space onto a paracompact (collectionwise normal, metacompact,...) space respectively. Moreover, we study some of the strengths and weaknesses of these notions.
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