On functionally θ-normal spaces

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 ≡ 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.

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 (≡ k-normal) space
• Almost normal space
• δ-normal space
• δ-normally separated