مهرداد کاروان جهرمی

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.