| In this thesis we mainly discuss the properties of "hereditary" and the properties under the project mappings of a special network called "weak base", which lies between the network and the topological base. And we also give some counterexamples within the Arens space S2. Since the structures of networks are more delicate and more variable, from 1960s, the investigation of generalized metric spaces is an active direction of General Topology all the time. The topology scholars have made all sorts of restricts to various networks. Then many classes of generalized metric spaces were draw in and studied.In this case, the concept of weak base was introduced by Arhangel'skii, and then many topology scholars investigated in this realm and got many good results such as weak base is "hereditary" with respect to closed or open subspace. But these topology scholars mainly paid their attentions to generalized metric spaces which have point countable weak bases, such as g-first countable, g-second countable, g-metrizable etc. There are few results for the weak base itself, especially the properties of "hereditary" and the properties of the Cartesian products.On these premise, we mainly discuss these properties of weak bases and got the following results:(l)weak base is "hereditary" with respect to k-subspace;(2)the weak base B is "hereditary" to any subspace of X if and only if for any x ∈ X, for any P ∈ Bx, x ∈ P°;(3)A is a subset of X, if for any x ∈ A satisfied x ∈ A° or for any Px ∈ Bx, x ∈ intA(Px∩A), then the weak base B is "hereditary" to A.(4)A is a subset of X, x is a nonisolate point of A, if there exist a. P ∈ Bx satisfied P ∩ A = {x}{or equivalent P ∩ A\{x} is a closed subset of A), then B is not "hereditary" to A.(5)B = ∪{Bx,y: x ∈ X, y∈ Y} is a weak base of the product space X ×Y, then P = ∪{Px, x ∈ X}, where Px= ∪y∈Y{p(B) : B ∈ Bx,y}, is not a weak base of X in general. But if we fix a point y0∈ Y, then P = ∪Px, where Px = {p{B) : B ∈ Bx,y0 is a weak base of X. |
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