We introduce new types of homogeneity, namely: locally homogeneity and closed homogeneity .Several results are included discussing some relations between these types and the old ones. Some characterization and decomposition theorems are obtained. Relevant examples and counterexamples are discussed throughout this paper. | Turk J Math 24 (2000) , 335 – 344. ¨ ITAK ˙ c TUB New and Old Types of Homogeneity Ali Ahmad Fora∗ Abstract We introduce new types of homogeneity ; namely : locally homogeneity and closed homogeneity .Several results are included discussing some relations between these types and the old ones. Some characterization and decomposition theorems are obtained. Relevant examples and counterexamples are discussed throughout this paper. Key Words: Homogeneity, countable dense homogeneity, strong local homogeneity, representable, n-homogeneity, weakly n-homogeneity. 1. Introduction If X is a topological space, then H(X) denotes the group of all autohomeomorphisms on X. Recall that a topological space X is weakly n-homogeneous (n ∈ N ) if for any A and B two n-element subsets of X, there is a homeomorphism h ∈ H(X) such that h(A) = B. A space X is n-homogeneous means that if A = {a1 , ., an} and B = {b1 , ., bn} are two n element subsets of X then there is a homeomorphism h ∈ H(X) such that h(ai ) = bi for all i = 1, ., n. A space X is called homogeneous if it is 1-homogeneous ( equivalently, if it is weakly 1-homogeneous). Let ∼ be the relation defined on X by x ∼ y if there is an h ∈ H(X) such that h(x) = y. This relation turns out to be an equivalence relation 1991 Mathematics Subject Classification: 54B, 54C. author is grateful to Yarmouk University for granting him a sabbatical leave in which this ∗ The research is accomplished. 335 FORA on X whose equivalence classes Cx will be called homogeneous components determined by x ∈ X. Cx is indeed a homogeneous subspace of X. It is clear that X is homogeneous if and only if it has only one homogeneous component. The connected component of X determined by x will be denoted by Kx while the quasi-component will be denoted by Qx . The homogeneity concept was introduced by W. Sierpinski [11] in 1920. Seven years earlier, . Brouwer [3] in his development of dimension theory had shown that if A and B are two .
