Base - Covering The Sub-compact Space, The Space Of Rank And Weak Base

This thesis consists of three parts: (i) base-cover metacompactness and base-familymetacompactness of products, (ii) the rank of the diagonal and the rank of the space and(iii) some properties on weak bases.In chapter 1, we introduce some notations and well known results about this thesis. Wealso list our work in this chapter.In chapter 2, we discuss base-cover metacompactness and base-family metacompactnessof products. Base-cover metacompact spaces and base-family metacompact spaces are insense of Popvassilev. In [46] and [47], Popvassilev posed:(a)[46, Question 3.4] If X is base-cover metacompact and Y is compact or metrizable,is X×Y base-cover metacompact? Is S×(ω+ 1) base-cover metacompact, where S is theSorgenfrey line?(b)[47, Question 3.7] What can we say a (regular) T2 space X if X×(ω+1) is base-familymetacompact?Answering these two questions, we prove:(2.1) The product of the Michael line (or the Sorgenfrey line) andω1+1 is not base-covermetacompact, whereω1 + 1 has the order topology.(2.2) The product of a base-cover metacompact Lindel¨of space and a compact metrizablespace is base-cover metacompact.(2.3) The product of two base-family metacompact spaces is base-family metacompact.Item (2.1) above shows that products of base-cover metacompact spaces and compactspaces need not be base-cover metacompact. It follows from (2.2) that the productS×(ω+1) of the Sorgenfrey line S andω+1 is base-cover metacompact. And (2.3) impliesthat X×(ω+ 1) is base-family metacompact if and only if X is base-family metacompact.In chapter 3, we investigate the rank of the diagonal and the rank of the space.Arhangel'ski?? and Buzyakova defined the rank of the diagonal and the rank of the spacewhich lie between Gδ-diagonal and submetrizability. They showed that Mrowka spaceφ(N)has a diagonal of the rank exactly 2. Then they constructed a Tychono? Moore space Zthat is separable, non-submetrizable, and has a diagonal of the rank exactly 3. In [4], theyalso gave the following conjecture. Conjecture: For each n∈ω, there is a Tychono? space Xn with a rank n-diagonal that isnot a rank (n + 1)-diagonal.In this chapter, we give an example of non-normal Tychono? Moore space that has adiagonal of the rank exactly 4.In chapter 4, we consider some properties of a special network called weak base, whichis between the network and the topological base. And we also give some counterexamplesusing Arens space S2. The concept of weak bases was proposed by Arhangel'skii, and thenmany topologists investigated in this realm and done much work such as weak bases arehereditary with respect to open subspaces. Liu proved weak bases are hereditary withrespect to closed subspaces [24, Lemma 2.1.] . But most of these topologists concentratedon generalized metric spaces with point countable weak bases, such as g-first countable, g-second countable, and g-metrizable. There are few results on the weak base itself, especiallythe properties of hereditary and the properties of the Cartesian products. In this chapter,we discuss these properties of weak bases and get the following results:(4.1)weak bases are hereditary with respect to k-subspaces.(4.2)the weak base B is hereditary to each subspace of X if and only if x∈intX(P),for each x∈X, and P∈Bx.(4.3)Let A be a subset of X. The weak base B is hereditary with respect to A, if foreach x∈A, either x∈intX(A), or x∈intA(Px∩A) for each Px∈Bx.(4.4)Let B = {Bx,y : x∈X,y∈Y } be 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 ingeneral. But if we fix a point y0∈Y , then P = {Px : x∈X}, where Px = {p(B) : B∈Bx,y0} is a weak base of X.(4.5)Let B = x∈X Bx and P = x∈X Px be weak bases on X, then:(i) B∧P = x∈X Bx∧Px is a weak base if X is a sequential space;(ii) B∨P = x∈X Bx∨Px is a weak base.And the assumption that X is a sequential space is necessary in (i).