The Generalization Of (?)₀-Weak Base And Relared Results

This dissertation is composed of two parts. The first part is the study of (?)₀ - weak base and some characterizations of (?)₀ - weak base. Discussions are also on relations between point-countable (?)₀ -weak base and sn-metric spaces. The second part gives a new concept which is called (?)₀ - sn - metric spaces, it gives some new characterizations.The main results are as following:Results 1 (Lemma 2.3) If B is a point-countable (?)₀-weak base, thenβis a cs^* -network. (?)₀ - weak base is open, closed hereditary.Results 2 (Theorem 2.2) The following are equivalent for a space X:(1) X has aσ-discrete (?)₀-weakbase;(2) X has aσ-locally finite (?)₀-weak base;(3) X is an (?)₀-weakly first countable and (?) -space.Results 3 (Theorem 3.4) Point-countable and (?)₀-weak base is equivalent to a sequential space with a point-countable (?)₀-sn- network and (?)₀ - sn - network is hereditary.Results 4 (Theorem 3.7) The following are equivalent for a space X:(1) X has aσ-discrete (?)₀ - sn - network;(2) X is an (?)₀ - sn - metric space;(3) X is an (?)₀-sn-weakly first-countable and (?)-space.Results 5 (Theorem 3.11) The following are equivalent for aspace X:(1) X is a sequentially quotient, countable-to-one image of a separable metric space; (2)X is an (?)₀-sn- weakly first-countable and a sequentially quotient image of a separable metric space;(3)X has countable (?)₀ - sn - network;(4)X is an (?)₀-sn-weakly first-countable and (?)₀-space.