基—覆盖亚紧空间、空间的秩和弱基
摘要
本文讨论了以下三个方面:(i)基?覆盖亚紧空间和基?集族亚紧空间的乘积性;(ii)开覆盖序列的秩和空间的秩以及(iii)弱基的基本性质.
在第一章,我们介绍了本文的背景、相关知识和主要结果,并对本文所用的符号加以说明.
在第二章中,我们讨论了基?覆盖亚紧空间和基?集族亚紧空间. Popvassilev在[47]中给出了这两类空间的定义,并研究了它们的遗传性和乘积性,得到一些结果,同时提出了许多问题.本文主要讨论了如下两个问题:
(a)[46,问题3.4]若X是基?覆盖亚紧空间,Y是紧空间或度量空间,那么X×Y是否是基?覆盖亚紧空间?又设S是Sorgenfrey直线,那么S×(ω+1)是否是基?覆盖亚紧空间?
(b)[47,问题3.7]设X是正则空间或Hausdor?空间,如果X×(ω+ 1)是基?集族亚紧空间,X应具有什么样的性质?通过对上述问题的研究我们得到如下结果:
(2.1) Michael直线(或Sorgenfrey直线)与ω1 + 1(赋予ω1 + 1线性序拓扑)的乘积不是基?覆盖亚紧空间.这表明基?覆盖亚紧空间与紧空间的乘积不一定是基?覆盖亚紧空间.
(2.2)基?覆盖亚紧的Lindel¨of空间与紧度量空间的乘积是基?覆盖亚紧空间.因为Sorgenfrey直线S是基?覆盖亚紧的Lindel¨of空间且ω+ 1是紧度量空间,所以S与ω+ 1的乘积是基?覆盖亚紧空间.
(2.3)两个基?集族亚紧空间的乘积仍然是基?集族亚紧空间.因此,X×(ω+1)是基?集族亚紧空间当且仅当X是基?集族亚紧空间.(2.1)和(2.2)回答了上面的问题(a)中除了Y是度量空间的问题;(2.3)则完全回答了问题(b).同时,它们还推广了[47,定理2.4].
在第三章中,我们讨论了开覆盖序列的秩和空间的秩.在[4]中, Arhangel’ski??和Buzyakova在Gδ对角线和次可度量的基础上给出了开覆盖序列的秩和空间的秩的定义,同时指出Mrowka空间φ(N)是秩?2不是秩?3的.随后他们又给出了一个可分的Tychono?Moore空间Z满足Z是秩?3不是秩?4的.在[4]中,他们还给出一个猜想.猜想:对每一个自然数n,都存在一个Tychono?空间Xn满足Xn是秩?n不是秩?(n + 1)的.基于上述,本章给出了一个非正规的Tychono? Moore空间Z4满足Z4是秩?4不是秩?5的.
在第四章中,我们研究了位于网和拓扑基之间的一种特殊的网——弱基的遗传性和它在投影映射下的性质,并以Arens空间S2为例对一些不成立的命题给出了反例.Arhangel’ski??(1966年)引进了弱基的概念,揭开了对弱基的研究.由定义我们容易得到弱基是开遗传的;刘川证明弱基是闭遗传的[24,引理2.1.].然而后来的拓扑学家主要在广义度量空间内研究具有某些点可数性质弱基的空间,比如g?第一可数空间、g?第二可数空间、g?可度量空间等.对弱基本身的研究却很少,甚至对弱基最基本的遗传性,乘积性,是哪些映射的不变量和逆不变量等问题还没有很好的答案.
这一章,我们对拓扑空间弱基的遗传性和它在投影映射下的性质进行了研究和探索,主要结果如下:
(4.1)弱基对k?子空间是遗传的.
(4.2)弱基B对X的任意一个子空间遗传当且仅当对于任意x∈X,P∈Bx,x∈intX(P).
(4.3)设A是X的一个子空间,如果对于任意x∈A满足x∈intX(A)或者对于任意Px∈Bx,x∈intA(Px∩A),则弱基B对A遗传.
(4.4)设B = {Bx,y : x∈X,y∈Y }是乘积空间X×Y的一个弱基,则P = {Px :x∈X}不一定是X的弱基,其中Px = y∈Y {p(B) : B∈Bx,y} .但是我们固定一个点y0∈Y ,则P = {Px : x∈X}是X的弱基,其中Px = {p(B) : B∈Bx,y0}.
(4.5)设B = {Bx : x∈X}和P = {Px : x∈X}都是X的弱基,则:
(i)如果X是序列空间,那么B∧P = {Bx∧Px : x∈X}是X的弱基;
(ii)B∨P = {Bx∨Px : x∈X}是X的弱基.且(i)中X是序列空间的假设不能去掉.
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).
在第一章,我们介绍了本文的背景、相关知识和主要结果,并对本文所用的符号加以说明.
在第二章中,我们讨论了基?覆盖亚紧空间和基?集族亚紧空间. Popvassilev在[47]中给出了这两类空间的定义,并研究了它们的遗传性和乘积性,得到一些结果,同时提出了许多问题.本文主要讨论了如下两个问题:
(a)[46,问题3.4]若X是基?覆盖亚紧空间,Y是紧空间或度量空间,那么X×Y是否是基?覆盖亚紧空间?又设S是Sorgenfrey直线,那么S×(ω+1)是否是基?覆盖亚紧空间?
(b)[47,问题3.7]设X是正则空间或Hausdor?空间,如果X×(ω+ 1)是基?集族亚紧空间,X应具有什么样的性质?通过对上述问题的研究我们得到如下结果:
(2.1) Michael直线(或Sorgenfrey直线)与ω1 + 1(赋予ω1 + 1线性序拓扑)的乘积不是基?覆盖亚紧空间.这表明基?覆盖亚紧空间与紧空间的乘积不一定是基?覆盖亚紧空间.
(2.2)基?覆盖亚紧的Lindel¨of空间与紧度量空间的乘积是基?覆盖亚紧空间.因为Sorgenfrey直线S是基?覆盖亚紧的Lindel¨of空间且ω+ 1是紧度量空间,所以S与ω+ 1的乘积是基?覆盖亚紧空间.
(2.3)两个基?集族亚紧空间的乘积仍然是基?集族亚紧空间.因此,X×(ω+1)是基?集族亚紧空间当且仅当X是基?集族亚紧空间.(2.1)和(2.2)回答了上面的问题(a)中除了Y是度量空间的问题;(2.3)则完全回答了问题(b).同时,它们还推广了[47,定理2.4].
在第三章中,我们讨论了开覆盖序列的秩和空间的秩.在[4]中, Arhangel’ski??和Buzyakova在Gδ对角线和次可度量的基础上给出了开覆盖序列的秩和空间的秩的定义,同时指出Mrowka空间φ(N)是秩?2不是秩?3的.随后他们又给出了一个可分的Tychono?Moore空间Z满足Z是秩?3不是秩?4的.在[4]中,他们还给出一个猜想.猜想:对每一个自然数n,都存在一个Tychono?空间Xn满足Xn是秩?n不是秩?(n + 1)的.基于上述,本章给出了一个非正规的Tychono? Moore空间Z4满足Z4是秩?4不是秩?5的.
在第四章中,我们研究了位于网和拓扑基之间的一种特殊的网——弱基的遗传性和它在投影映射下的性质,并以Arens空间S2为例对一些不成立的命题给出了反例.Arhangel’ski??(1966年)引进了弱基的概念,揭开了对弱基的研究.由定义我们容易得到弱基是开遗传的;刘川证明弱基是闭遗传的[24,引理2.1.].然而后来的拓扑学家主要在广义度量空间内研究具有某些点可数性质弱基的空间,比如g?第一可数空间、g?第二可数空间、g?可度量空间等.对弱基本身的研究却很少,甚至对弱基最基本的遗传性,乘积性,是哪些映射的不变量和逆不变量等问题还没有很好的答案.
这一章,我们对拓扑空间弱基的遗传性和它在投影映射下的性质进行了研究和探索,主要结果如下:
(4.1)弱基对k?子空间是遗传的.
(4.2)弱基B对X的任意一个子空间遗传当且仅当对于任意x∈X,P∈Bx,x∈intX(P).
(4.3)设A是X的一个子空间,如果对于任意x∈A满足x∈intX(A)或者对于任意Px∈Bx,x∈intA(Px∩A),则弱基B对A遗传.
(4.4)设B = {Bx,y : x∈X,y∈Y }是乘积空间X×Y的一个弱基,则P = {Px :x∈X}不一定是X的弱基,其中Px = y∈Y {p(B) : B∈Bx,y} .但是我们固定一个点y0∈Y ,则P = {Px : x∈X}是X的弱基,其中Px = {p(B) : B∈Bx,y0}.
(4.5)设B = {Bx : x∈X}和P = {Px : x∈X}都是X的弱基,则:
(i)如果X是序列空间,那么B∧P = {Bx∧Px : x∈X}是X的弱基;
(ii)B∨P = {Bx∨Px : x∈X}是X的弱基.且(i)中X是序列空间的假设不能去掉.
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).
引文
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