| dc.description.abstract | A simple extension of a topology is defined by Norman Levine as follows: Let (X,T) be a topological space where T is some topology on X, and let A be a subset of X such that A is not in T. Then the topology T(A) = (U union (U' intersection A) | U, U' in T, U in T} is called a simple extension of T. Basic theorems and lemmas having to do with a simple extension of a topology, which were proved by Levine, are cited, and some examples of topological spaces with extended topologies are given. A concept of an extension of a topology by an arbitrary number of sets is introduced. Such an extension is denoted T[A_alpha] where G = {A_alpha |A_alpha in X, A_alpha not in T, alpha in A} is the collection of subsets of X by which T is extended. We show that an extension of the form T[A_alpha] is equivalent to a well-ordered succession of simple extensions. It is proved that if T_1 and T_2 are topologies for a set X and G_2 = {T_alpha | T_alpha not in T_1, T_alpha is a member of an open basis for T_2, alpha in N, then for every filter F on X the set C = A intersection B (where A is the set of convergence points of F with respect to T_1, B is the set of convergence points of F with respect to T_2) is the set of convergence points of F with respect to T_3 iff T_3 = T_1[T_alpha]. We generalize this theorem to the case in which we have an arbitrary number of topologies and extend one of these topologies by the members of the open bases for the other topologies. A necessary and sufficient condition is proved for T[A_alpha] = T(Union where alpha is in Lambda, A_alpha) where T is a topology for a set X and G = {A_alpha in X, A_alpha not in T, alpha in Lambda}. In addition, necessary and sufficient conditions are proved for T(A) = T(B) and also for T(A) subset of T(B) where A and B are subsets of X such that A, B not in T. Next we are concerned with preservation of topological properties under an extension of a topology. We prove necessary and sufficient conditions for connectedness, regularity, full normality, local compactness, and dimension n at each point of X to be preserved in (X, T(A)) if (X, T) is respectively connected, regular, fully normal, locally compact, and of dimension n at each point of X where T(A) is a simple extension of T. Furthermore, we give theorems and proofs, from C. J. R. Borges, of necessary and sufficient conditions for preserving complete regularity, normality, perfect normality, collectionwise normality, and metrizability in (X, T(A)) if (X,T) ii respectively completely regular, normal, perfectly normal, collectionwise normal, and metrizable where T(A) is a simple extension of T, Several instances are cited in which topological properties of (X,T(A)) may or may not be possessed by (X,T) where T(A) is a simple extension of T. If T_1 and T_2 are topologies for a set x. we consider the topology T acquired by extending T_1 by the members of an open basis for T_2 which do not belong to T_1. We show that if (X, T_1) is T_0, T_1, or T_2 then (X, T) is respectively T_0, T_1, or T_2. It is proved that (X, T) is regular or completely regular if both (X, T_1) and (X, T_2) are respectively regular or completely regular. We give an example in which (X, T_1) and (X, T_2) are both normal but (X, T) is non-normal, and another example in which (X, T_1) and (X, T_2) are both compact but (X, T) is non-compact. | |
