On functionally θ-normal spaces
Keywords:θ-closed (open) set, Regularly closed (open) set, Zero set, Regular Gδ-set, (weakly) (functionally) θ-normal space, (weakly) θ-regular space, Almost regular space, Mildly normal (â‰¡ k-normal) space, Almost normal space, δ-normal space, δ-normally separated
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 â‰¡ k-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.
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