A new generalization of normality called almost β-normality is introduced and studied which is a simultaneous generalization of almost normality and β-normality. A topological space is called almost β-normal if for every pair of disjoint closed sets A and B one of which is regularly closed, there exist disjoint open sets U
Abstract. The class of θ-compact spaces is introduced which properly contains the class of almost compact (generalized absolutely closed) spaces and is strictly contained in the class of quasicompact spaces. In the realm of almost regular spaces, the class of θ-compact spaces coincides with the class of nearly compact spaces. Moreover, an almost regular θ-compact space is mildly normal (= κ-normal). A θ-closed, θ-embedded subset of a θ-compact space is θ-compact and the product of two θ-compact space is θ-compact if one of them is compact. A (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). A space is compact if and only if it is θ-compact and θ-point paracompact.
Abstract. Characterizations of functionally θ-normal spaces including the one that of Urysohn's type lemma, are obtained. Interrelations among (functionally) θ-normal spaces and certain generalizations of normal spaces are discussed. It is shown that every almost regular (or mildly normal ≡ κ-normal) θ-normal space is functionally θ-normal. Moreover, it is shown that every almost regular weakly θ-normal space is mildly normal. A factorization of functionally θ-normal space is given. A Tietze's type theorem for weakly functionally θ-normal space is obtained. A variety of situations in mathematical literature wherein the spaces encountered are (functionally) θ-normal but not normal are illustrated.
2000
Abstract.Generalizations of normality, called (weakly) (functionally) ∆-normal spaces are introduced and their interrelation with some existing notions of normality is studied. ∆-regular spaces are introduced which is a generalization of seminormal, semiregular and θ-regular space. This leads to decompositions of normality in terms of ∆-regularity, seminormality and variants of ∆-normality.2000 AMS Classification: 54D10, 54D15, 54D20.
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