基于广义Zadeh函数的L-双拓扑空间的几类弱连通性

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基础拓扑学讲义尤承业答案

基础拓扑学讲义1. 引言拓扑学是数学的一个分支，研究的是空间中的集合如何存在和连接的学科。

在拓扑学中，我们关注的是集合之间的关系，而不是集合的具体性质。

本讲义将介绍拓扑学的基础知识和常见概念。

2. 基本概念2.1 集合在拓扑学中，一个集合是指由元素组成的无序对象的集合。

示例：- 集合 A = {1, 2, 3, 4}- 集合 B = {a, b, c}2.2 拓扑空间拓扑空间是指一个集合论中的空间，其中具有一组满足特定条件的子集。

示例：- 欧几里得空间- 流形空间- 度量空间2.3 拓扑结构拓扑结构是指在拓扑空间中定义的一组特殊集合的集合，它满足特定的公理。

示例：- 开集- 闭集- 连通集- 紧集3. 拓扑学的基本性质3.1 连通性在拓扑学中，连通性是指一个拓扑空间中不存在将空间分为两个或更多部分的拓扑属性。

示例：- 实数集 R 是一个连通集- 平面上的一个圆形是一个连通集3.2 完备性在拓扑学中，完备性是指拓扑空间中的每个柯西序列都有一个收敛的极限。

示例：- 实数集 R 是一个完备的度量空间3.3 紧集在拓扑学中，一个集合被称为紧集，如果它的每个开覆盖都具有有限子覆盖。

示例：- 闭区间 [0, 1] 是一个紧集4. 拓扑学的应用拓扑学在各个领域都有广泛的应用，包括物理学、生物学、计算机科学等。

示例：- 电路板设计中的连接问题- 生物分子的空间构象研究- 网络拓扑结构的分析与优化5. 总结本讲义介绍了拓扑学的基本概念、拓扑结构和其应用。

拓扑学作为一门重要的数学学科，对于理解和描述空间的性质具有重要的作用。

希望通过本讲义的学习，能够对拓扑学有一个初步了解，并能够应用于实际问题中。

注意：本讲义只是拓扑学的入门讲义，如果想深入学习拓扑学，请参考相关的高级教材和论文。

低维拓扑简介

考研论坛»数学»低维拓扑knight51发表于2005-7-28 08:34低维拓扑<P>下面说说低维拓扑的内容：低维拓扑是微分拓扑的一部分，主要研究3，4维流形与纽结理论。

又叫几何拓扑。

主要以代数拓扑与微分拓扑为工具。

它与微分几何和动力系统关系密切。

国外搞这个方向的也几乎都搞微分几何和动力系统。

我国这个方向北大最牛，美国是伯克利和普林斯顿最牛。

比起代数几何来，它比较好入门。

初学者只需要代数拓扑，微分拓扑，黎曼几何的知识就行了。

美国这方面比较牛，几乎每个搞基础数学研究的都会低维拓扑。

</P><DIV class=postcolor>纠正一下上面的错误，美国也不是每个搞基础数的都精通低维拓扑，而是懂一些低维拓扑的知识。

如果入门后还想更加深入了解它，那还需要读一些双曲几何和拓扑动力系统的书。

</DIV><DIV class=postcolor>下面介绍一下这方面的牛人：Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplaybetween analysis, topology and geometry.Wall [8] goes on to describe Thurston's work in more detail:-The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).</DIV>第2个牛人：Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaréconjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.Besides these results about closed simply connected four manifolds, Freedman also proved:(a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R.(b) There is a nonsmoothable closed four manifold.© The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:-My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.<DIV class=postcolor>介绍第3个牛人：Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case.In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:-Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold.Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-(1) Differential geometry of holomorphic vector bundles.(2) Applications of gauge theory to 4-manifold topology.and he relates his contribution to that of many others in the field.Donaldson's work in summed up by R Stern in [6]:-In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds</DIV><DIV class=postcolor>下面continue介绍第4个牛人：Robion Kirby。

弦理论中的时空拓扑与CP破缺

弦理论中的时空拓扑与CP破缺在现代理论物理学中，弦理论被认为是追求统一描述自然界基本粒子与引力相互作用的最有希望的理论之一。

作为一种理论框架，弦理论涉及到多元素、复杂的数学结构，其中时空拓扑和CP破缺是两个重要而又深奥的概念。

一、时空拓扑在弦理论中，时空被视为一种弯曲、扭曲的几何结构。

时空拓扑是指时空的形状和连接性质。

与经典的几何学不同，时空拓扑关注的是形变和连通性的特征，而不依赖于具体的度量。

在弦理论中，时空的拓扑结构对于理解粒子的性质和相互作用非常关键。

时空拓扑的一个重要性质是维度。

在经典物理学中，我们熟知的是四维时空。

然而，弦理论中并不局限于四维，它提供了一种统一的框架来描述任意维度的时空。

这种超越我们传统经验的维度观念给予了研究者在理论空间中探索新奇物理现象的机会。

另一个重要的时空拓扑性质是曲率。

时空在弦理论中可以是平坦的，也可以是弯曲的。

根据爱因斯坦的广义相对论，我们知道质量和能量会弯曲时空，形成引力。

而在弦理论中，质量和能量的分布也会对时空产生影响，进而影响弦的运动和相互作用。

因此，时空拓扑的形状和曲率不仅与物质的分布有关，还与弦的本质密切相关。

二、CP破缺CP破缺是指物理理论中的CP对称性在某些特定条件下失效的现象。

CP对称性是指物理定律在对于粒子的粒子-反粒子变换（C）以及时间（P）翻转下保持不变。

而在一些物理现象中，我们观察到对于CP变换而言，粒子的行为与其反粒子并不完全相同，从而表现出了CP破缺的现象。

在弦理论中，CP破缺是一个重要的研究课题。

在标准模型下，CP破缺被观测到，并与强子物理中的强子-反强子振荡现象等相关。

而弦理论作为更高级的理论，可以为CP破缺提供一种更深入的解释。

通过弦理论的研究，我们可以深入探索CP破缺的本质，并寻找与之相关的新物理现象。

三、弦理论中的时空拓扑与CP破缺的关系时空拓扑和CP破缺在弦理论中存在着密切的联系。

时空拓扑的形状和曲率对于物质的性质和相互作用至关重要，而CP破缺则与物质的对称性有关。

基础拓扑学讲义尤承业版

基础拓扑学讲义尤承业版第一章绪论1.1 拓扑学的定义与发展拓扑学是数学的一个分支，研究空间中的性质在连续变形下的不变性。

本章介绍了拓扑学的定义、发展历程以及基本概念。

1.2 拓扑学的基本概念本节介绍了拓扑学中的一些基本概念，包括集合、点集、邻域、开集、闭集等。

并且详细解释了它们的定义和性质。

第二章拓扑空间2.1 拓扑空间的定义本节介绍了拓扑空间的定义，即一个集合和一个定义在该集合上的拓扑结构构成的数学结构。

2.2 拓扑空间的基本性质本节介绍了拓扑空间的基本性质，包括空间的连通性、紧致性、分离公理等。

并且给出了相应的定义和定理。

第三章连续映射与同胚3.1 连续映射的定义本节介绍了连续映射的定义，即在拓扑空间之间保持连续性的映射。

3.2 同胚的定义与性质本节介绍了同胚的定义，即两个拓扑空间之间存在一个双射映射，并且该映射和其逆映射都是连续映射。

第四章拓扑基与拓扑生成4.1 拓扑基的定义与性质本节介绍了拓扑基的定义，即一个拓扑空间中的开集可以由拓扑基中的元素表示。

并且给出了拓扑基的一些性质。

4.2 拓扑生成的定义与性质本节介绍了拓扑生成的定义，即一个集合可以通过某些子集的交、并、补运算生成一个拓扑空间。

第五章度量空间与距离5.1 度量空间的定义与性质本节介绍了度量空间的定义，即一个集合中的元素可以通过距离函数相互比较。

5.2 距离函数的性质本节介绍了距离函数的性质，包括非负性、对称性、三角不等式等。

第六章完备性与紧致性6.1 完备性的定义与性质本节介绍了完备性的定义，即度量空间中的某个子集的极限点都在该集合内。

6.2 紧致性的定义与性质本节介绍了紧致性的定义，即一个拓扑空间中的任意开覆盖都存在有限子覆盖。

第七章分离公理7.1 Hausdorff空间的定义与性质本节介绍了Hausdorff空间的定义，即一个拓扑空间中的任意两个不同点都存在不相交的邻域。

7.2 正则空间和完全正则空间本节介绍了正则空间和完全正则空间的定义，以及它们与Hausdorff空间的关系。

吴济民_反德西特空间

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3. 连通性: 若 X 不能表示为两个不相交的非空开子集的并集 , 则称 X 为连通的.
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变分形式1,2

§1. 二次函数的极值在n维欧氏空间Rn中引入向量和矩阵记号:
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从本章开始, 将逐步涉及用有限元方法求解微分方程的边值问题，这种方法属于变分法的范畴, 是古典的变分法与分片多项式插值相结合的产物。这种结合使得有限元方法不仅保持了变分法的优点, 而且可以通过一种标准的过程在电子计算机上实现, 从而弥补了古典变分方法的不足。 “有限元方法” (Finite Element Method) 这一术语首次出现于R. W. Clough 1960年9月发表的一篇讨论平面弹性力学问题的力学论文中。然而, 这种方法的思想却早已有之。故“谁首先”、“在何时” 提出有限元方法的问题在数学家、物理学家和工程师之间就有三种不同的答案。
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完全的线性赋范的内积空间hilbert空间是banach空间的特例它具有较banach空间更丰富的性质现在回过来看二次泛ju假定显然应要求所谓完全是指cauchy收敛定理于l为基本序列则必有并记为lim然而仅限于此还是不够的因为为此我们引进广义导数广义微商概念

泛函分析基本定理证明

泛函分析是数学中的一门重要学科，它研究的是无限维空间中的函数和函数列的性质。

在泛函分析中，有一些基本定理被广泛应用于实际问题的解决中。

本文将证明泛函分析中的两个基本定理：闭线性子空间的补空间存在性和开映射定理。

首先，我们来证明闭线性子空空间的补空间存在性定理。

设X是一个Banach空间，Y是它的一个闭线性子空间。

我们定义X的柯西序列为{xn}，它满足对于任意的ε>0，存在正整数N，当m,n>N时有||xm-xn||<ε。

现在，我们取X的一个柯西序列{xn}，它在Y中取值为0。

我们定义序列{yn}为xn-x，其中x是Y的一个元素。

显然，对于任意的ε>0，当m,n>N时有||ym-yn||=||xm-xn-(x-x)||<ε，因此{yn}是Y的一个柯西序列。

由于Y是一个闭空间，所以{yn}收敛于Y中的一个元素y，即存在一个元素y∈Y，使得yn→y。

现在我们来证明y是X的一个元素。

由于Y是一个线性空间，我们知道对于任意的a∈F，b∈Y，ax+b∈Y。

对于任意的正整数m，有xm-yn=ym-yn+xm-yn=ym-x+xm-yn∈Y。

因此，{yn}是X的一个柯西序列。

由于X是一个Banach空间，所以{yn}收敛于X中的一个元素x0，即存在一个元素x0∈X，使得yn→x0。

现在我们来证明x0=y，即yn→y。

由于yn=yn+xn-x=y+xn-x，当n→∞时有yn→y，因此y=x0。

因此，我们证明了闭线性子空间的补空间存在性。

接下来我们来证明开映射定理。

设X和Y是两个Banach空间，T:X→Y是一个线性映射，并且存在正数M，使得对于任意的x∈X，有||Tx||≤M||x||。

我们要证明T是一个开映射，即T(U)是Y中的一个开集，其中U是X中的一个开集。

设x0∈U，由于U是一个开集，存在一个正数ε，使得B(x0,ε)={x∈X:||x-x0||<ε}⊆U，其中B(x0,ε)表示以x0为中心，ε为半径的开球。

基于改进SOGI-PLL电网基波正序分量同步方法

式中，％为锁相环输出的正序分量相位角。
图 1 传统二阶广义积分器结构图 Fig. 1 Diagram of traditional SOGI structure
图中输入信号为v» ，输出信号为和原信号相同的信号v» , 及移相 90°的信号
由图 1 得到传统的S O G I 传递函数为
\⑷ 崖
= U P„ 0 + u „ s\n{-2cot+ (/>)
⑶
式中， 0 = A —供。， 0 为负序电压分量相位角。
设和％中的直流分量和交流分量分别为
Ud~dc > Ud-act Uq-dc-> Uq~〇c i 贝丨j舍—
k = Ud-ac+ ^-dc
(4)
Ud-dc
UP
(5)
Uq-dc
0
V a c " U ncos(-2cot+ </>)
U ns\n{-2cot+ (/>)
(6)
电压不对称故障下，而轴电压分量中混有二倍频的交流分量，因此单同步坐标系锁相环无法实现准确锁相和电压基波信号的正负序分离[14]。
3 基于改进 SOGI-P L L 的电网基波正序分量同步方法
者，Emaik 2 7 3 8 2 4 3 1 3 @ q q . c o m ) ; 王安娜(1956-),女，辽宁鞍山人，博士，教授，主要从事智能电网、模式识别、故障诊断等方面的教学与科研工作。
第6期
唐爱博等：基于改进SOGI-P L L 电网基波正序分量同步方法
•1137 •
相位、频率信息。在电网发生不对称故障时，负序分量的波动会影响锁相结果的准确性，为实现精确锁相，必须准确跟踪基波电压信号的正序分量 [5,6]。文献 [7]提出基于频率前馈的新型三相锁相环结构来实现正序分量跟踪，在却坐标系下利用双二阶广义积分器进行瞬时对称分量运算分离出基波电压信号的正负序分量。但瞬时对称分量运算也会减缓系统的响应速度，同时二阶广义积分器对高次谐波具有较好的抑制效果，对低次谐波的抑制效果较差，当混有直流分量时也无法完成电压信号的准确检测 [8]。文献 [9〜11]利用特定频率的陷波器来消除电网电压不对称时而轴分量中的2倍工频波动量，但陷波器参数选取较复杂，且没有考虑直流分量和频率相位偏移的影响。文献[12，13]利用多级联模块来消除却坐标系下全谐波分量对锁相结果的影响，但多级联结构增加了系统延时，减缓了响应速度。

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Ｊｕｎ．２０１５
文章编号
１０００— ５２６９（２０１５）０３— ０００１ — ０３
ＤＯＩ：１０．１５９５８／ｊ．ｃｎｋｉ．ｇｄｘｂｚｒｂ．２０１５．０３．Ｏ１
基于广义Ｚａｄｅｈ函数的一双拓扑空间的几类弱连通性
一
７］基础上，借助广义ｚａｄｅｈ通性，ＩＦＰＯ一０连通性及ＰＯ一０连通性等问题，得出这三类连通性是０一
双弱同胚不变性。
０连通的。设（Ｌ，１，￣ＪＯ２）是一掘，Ｄ ∈Ｌ，
—
定义１．３
设（Ｌ，）是Ｌ—ｔｓ，ｓ∈Ｌ一｛１｝，Ｄ
∈Ｌ，Ｄ称为０一０连通的，如果不存在Ｇ，日∈ Ｃａ），使得Ｄ《Ｇ，Ｄ《Ｈ，Ｄ≤ ＧＶＨ且Ｄ八Ｇ八
日≤Ｃ；（Ｌ， ∞）称为０一０连通的，如果是０
Ａ，Ｂ ∈Ｌ，Ａ，Ｂ称为０一０隔离的，如果Ａ＾Ｂ≤ Ｃ且＾ ≤Ｃ，其中Ｃ表示取常值ｓ的Ｆ集。
面，近年来，国内外学者对其性质进行了充分讨论，并将其推广到ｆｕｚｚｙ拓扑空间和．拓扑空间中进行研究，见文［１—３］，但至今为止，有关一双拓扑空间中的０一０连通性研究尚不完善。本文在文［１
中是ＷＰＯ一０隔离的，ＶＢ＝Ｄ，Ａ《Ｃ，Ｂ《
是０一闭集，当Ａ是０一闭集时，称Ａ是０．开集，Ａ的补集的０一闭包的补集称为Ａ的０一内部，记为Ａ；，
显然ａｏ＝＾｛Ｂ ∈Ｒ（）ＩＡ ≤ Ｂ｝，
．
拓扑空间（Ｌ， ∞ ）（ｉ＝１，２）中Ａ的０一闭包。
设（Ｌ，】， ∞ ２）是￡一，Ｄ ∈Ｌ，
定义１．６
并称Ａ是Ａ的０一闭包，如果Ａ＝ａｏ，则称Ａ
如果不存在Ａ，Ｂ∈Ｌ，使得Ａ，曰在（ ∥，， ∞ ）
第３２卷第３期２０１５年６月
贵州大学学报（自然科学版）ＪｏｕｎａｒｌｏｆＧｕｉｚｈｏｕＵｎｉｖｅｒｓｉｔｙ（ＮａｔｕｒａｌＳｃｉｅｎｃｅｓ）
Ｖｏ１．３２Ｎｏ．３
姜金平，王小霞，赵宁宁，李萍
（延安大学数学与计算机科学学院，陕西延安７１６０００）
摘
要：借助广义Ｚａｄｅｈ函数，研究一双拓扑空间中ＢＯ一０连通性，ＩＶＰＯ一０连通性及ＰＯ，一
连通性的有关性质，得出这三类连通性是一双弱同胚不变性这一结论。
关键词：Ｌ一双拓扑空间；ＢＯ一０连通性；ＷＰＯ一０连通性；ＰＯ，一０连通性；广义Ｚａｄｅｈ函数中图分类号：０１８９．１３文献标识码：Ａ
双拓扑空间理论是拓扑学研究的一个重要方
１预备知识
定义１．１设（， ∞）为Ｌ — ｔｓ，称 ∈ Ｃｏｐｒ（Ｌ）是Ａ ∈ Ｌ的０．附着点，若ＶＰ ∈ Ｒ（），有Ａ《Ｐ，Ａ的所有０．附着点之并记作
，
（Ｌ，ｃ￡，，ｃＩＪ：）中是ＰＯ一０隔离的。这里Ａ表示
称为Ｌ一双拓扑空间，简记为Ｌ—ｂｔｓ。
若Ａ。八Ｂ ≤ Ｃ，Ａ八
≤Ｃ或Ａ＾ ≤
Ｃ，Ａ人Ｂ， ≤Ｃ，则称Ａ，在（，Ｏ０ｌ， ∞ ２）中是ＩＶＰＯ一０隔离的；若两式同时成立，则称，日在
定义１．５ ‘ ６设（，】， ∞ ２）是一
，
ｔｓ，Ａ一，Ａ。与Ａ分别表示Ａ ∈Ｌ的闭包，内部，伪设 ∞，，都是上一拓扑，则（Ｌ，， ∞ ）
，Ａ，Ｂｆｉ：
补，文中未出现的概念及记号参见文［８，９］。
定义１．４
设是，格，即具有逆序对合对应的完全分配格，为非空分明集， ∥ 表示上的全体．集，中的最大元与最小元分别记作１和０。设为 ∥ 上一拓扑，则（Ｌ，）称为Ｌ一拓扑空间，简记为
一
若Ｄ既是（Ｌ，。）中的０一０连通集，也是（Ｌ， ∞ ）中的０一０连通集，则称Ｄ是（Ｌ，。，）中
的ＢＯ一０连通集，特别地，当１是ＢＯ一０连通集
时，称（Ｌ，，）为ＢＯ一０连通空间。