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ABSTRACT. An ideal on a set X is a nonempty collection of subsets of X closed under the operations of subset and finite union. Given a topological space X and an ideal 2" of subsets of X, X is defined to be 2"-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all of X except for a set in 2". Basic results are investigated, particularly with regard to the 2"-paracompactness of two associated topologies generated by sets of the form Uwhere U is open and E 2" and U {UIU is open and U-A E 2", for some open set A}. Preservation of 2"-paracompactness by functions, subsets, and products is investigated. Important special cases of 2"-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya ["On mparacompact spaces", Math. Ann., 181 (1969), 119-133]

Two relationships considered by Weston [1] for a pair of topologies on a set X are translated to a function setting. An attempt to characterize the two resulting types of functions leads to new characterizations of weak continuity and almost continuity. After showing that weak continuity and almost continuity are independent, interrelationships are sought. This leads to the definition of subweak continuity and a new characterization for almost openness. Finally, several published results are strengthened or slightly extended

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By considering lower density operators and their induced topologies in a general setting, some results of S. Scheinberg and E. Lazarow et al. are unified and generalized.It is also shown that every u-finite complete measure space (X, A', m ) has a lower density operator and that every such operator induces a topology makingXa category measure space in the sense of J. C. Oxtoby, except that the measure need not be finite. One consequence is that category u-finite measure spaces must have the countable chain condition. Also, for every topological space (X, T), there is a lower density operator on the u-field of sets having the property of Baire (relative to the u-ideal of meager sets). Further, in both the "measure" and "category" contexts, all induced lower density topologies have simple form. Finally, it is shown that the deep.Pdensity operator on the u-field of subsets of the real line having the property of Baire is not a lower density operator. Mathematics Subject Classification Primary 28A05,54A10.Key words and phrases. Lower density operator, lower density topology, (u-)field, (a-)ideal, set having the property of Baire, meager set, nowhere dense set, (u-) finite measure space, category measure space, CCC (countable chain condition).