拓扑学

The students in a topology course should have some knowledge of mathematical analysis and abstract algebra.

拓扑学要求学生具备一些数学分析和抽象代数的基础。

From then on, more and more algebraic topologist become interested in this area.

从此，越来越多的代数拓扑学家对这一领域产生了兴趣。

At this time the subject of topology was known as analysis situs.

在这个时候，拓扑学被称做位置几何学。

At is time the subject of topology was known as analysis situs.

在这个 时候，拓扑学被称做位置几何学。

The modern physics development by to the concrete material embodiment inspection, moved towards to the abstract material abstract research, namely from specially moves towards general, its utilizes the mathematical instrument is also same, moves towards mathematics from physical mathematics physics, already developed the abstract algebra and the general analysis situs, from bad on the one hand said, the physicist because of lost the concrete material object to change confusedly, was at a loss how to proceed, from was good on the one hand says, the research technique from specially moved towards general, also caused its thinking mode to have the return, tended to the philosophy, this speaking of the physics, was one kind of significant crisis, but at the same time, Also is a brand-new development turning point.

现代物理学的发展已由对具体物质的具象考察，走向对抽象物质的抽象研究，即从特殊走向一般，其所运用的数学工具也一样，从物理的数学走向数学的物理，已经发展到抽象代数和一般拓扑学，从坏的一方面讲，物理学家因失去了具体的物质对象而变得迷茫，无所适从，从好的一方面讲，研究方法的从特殊走向一般，也使其思维方式发生了回归，趋向于哲学，这对物理学而言，是一种重大危机，但与此同时，也是一个全新的发展契机。

The linear system optimum control theory distributes one of parameter systembasic topics,in domains and so on aerospace,national defense,finance,communicationhas the widespread application prospect.simultaneously, because the research tech-nique involves to foundation mathematics theories and so on the functional analysis,analysis situs,geometry, algebra.

线性系统的最优控制理论是分布参数系统的基本课题之一，在航空航天、国防、金融、通讯等领域有着广泛的应用前景。同时，由于研究手法涉及到泛函分析、拓扑学、几何学、代数学等基础数学理论。

As we all known, with the founding of Euclidean geometry in ancient Greece, with the development of analytic geometry and other kinds of geometries, with F.Kline" s Erlanger program in 1872 and the new developments of geometry in 20th century such as topology and so on, man has developed their understand of geometry. On the other hand, Euclid formed geometry as a deductive system by using axiomatic theory for the first time. The content and method of geometry have dramatically changed, but the geometry curriculum has not changed correspondingly until the first strike from Kline and Perry" s appealing.

纵观几何学发展的历史，可以称得上波澜壮阔：一方面，从古希腊时代的欧氏综合几何，到近代解析几何等多种几何的发展，以及用变换的方法处理几何的埃尔朗根纲领，到20世纪拓扑学、高维空间理论等几何学的新发展，这一切都在不断丰富人们对几何学的认识；另一方面，从欧几里得第一次使用公理化方法把几何学组织成一个逻辑演绎体系，到罗巴切夫斯基非欧几何的发现，以及希尔伯特形式公理体系的建立，极大地发展了公理化思想方法，不管是几何学的内容还是方法都发生了质的飞跃。

In this paper,based on fundamental topology theories,the concept ofcomplete set is proposed and the closed ball model is constructed.

本文以拓扑学为基础，提出了完备集的概念，建立了拓扑关系的闭球模型。

This course will focus on some fundamental knowledge and methods of general topology and algebraic topology, including the theory of topological spaces, continuous maps, fundamental group, the theory of covering spaces and simplicial homology and so on.

本课程将着重介绍点集拓扑学和代数拓扑学的一些基本知识和基本方法，包括拓扑空间理论、连续映射、基本群、覆盖空间理论和单纯同调群理论等。

And its propositions are either childishly obvious （that is, until you try to prove them）or so difficult and abstract that not even a topologist can explain their intuitive meaning. But topology is no queerer than the physical world as we now interpret it.

拓扑学表面看来是一门奇特的学问；它探求的都是的难以些奇怪的难以置信的图形，它的定理要么是明显地带股孩子气，要么非常困难和抽象，即使一个拓扑学家也不可能凭知觉解释清它们的意义。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。