dc.description.abstract | A collection of sets {U( p) I p in R} is said to be symmetric if p in U(q) implies q in U(p) for every p, q in R. This thesis investigates the effects of this property on other known topological properties. For example, in a collection of symmetric sets, point finiteness is equivalent to star finiteness. Using this property a number of equivalent conditions for the star finite property, strong metrizability, metrizability and complete paracompactness are obtained. It is shown that the M_1 and M_3 spaces introduced by Cedar are metrizable if their bases are symmetric. It is shown that in the definition of the star finite property the word "open" used in referring to the refinement can be omitted if other suitable conditions are present, for example if the sets are symmetric neighborhoods. We show that a regular space R is strongly metrizable if and only if it has neighborhood bases {U_n(p) | n = 1, 2, . . . }, p in R, such that U_n = {U_n(p) | p in R} is star finite (locally finite, point finite, star countable, locally countable, point countable) and symmetric. Similarly we prove that a regular space R is completely paracompact if and only if every open covering of R has a refinement U' = Union from n=1 to infinity of U_n", U_n' = {U_n(p) | p in P_n subset of R}; Union from n=1 to infinity of P_n =R, which is a subcollection of U= Union from n=1 to infinity of U_n(p) | p in R}, where U_n is a star finite (locally n=l n n l finite, point finite, star countable, locally countable, point countable) collection of symmetric open sets. A theorem of Morita is generalized in arriving at conditions under which complete paracompactness is equivalent to the star finite property and to the Lindelof property. A new proof of the equivalence of the basic dimension concepts in strongly metrizable spaces is presented. A number of mapping theorems are proved, such as the invariance of the star finite property (strong metrizability, complete paracompactness) under an open continuous mapping if inverse images of points are connected. Two classes of generalized metric spaces, A_1 and A_2 spaces are defined and their properties studied. It is shown that they include the class of metrizable spaces and are contained in the class of collectionwise normal spaces. | |