simply connected

简明释义

单连通的

英英释义

例句

1.In topology, a space is called simply connected if it is path-connected and every loop can be continuously contracted to a point.

在拓扑学中，一个空间被称为简单连通，如果它是路径连通的，并且每个循环都可以连续收缩到一个点。

2.When studying complex functions, we often require the domain to be simply connected for certain theorems to hold.

在研究复函数时，我们通常要求定义域是简单连通的，以使某些定理成立。

3.The surface of a sphere is simply connected, meaning any loop on its surface can be shrunk to a single point without leaving the surface.

球体的表面是简单连通的，这意味着其表面上的任何循环都可以收缩到一个点，而不离开表面。

4.In contrast, a torus is not simply connected because there are loops that cannot be contracted to a point.

相比之下，圆环不是简单连通的，因为存在一些循环无法收缩到一个点。

5.A simply connected region in the complex plane allows for the application of Cauchy's integral theorem.

复平面中的一个简单连通区域允许应用柯西积分定理。

作文

In the field of topology, a fascinating branch of mathematics, the concept of simply connected plays a crucial role in understanding the properties of different spaces. A topological space is said to be simply connected if it is both path-connected and has no 'holes' in it. This means that any loop in the space can be continuously contracted to a single point without leaving the space. To illustrate this idea, consider a rubber band placed on a flat surface. If you draw a loop with the rubber band, you can easily shrink it down to a point without lifting it off the surface. Therefore, the flat surface is simply connected.On the other hand, imagine a torus, which is shaped like a doughnut. If you draw a loop around the hole of the torus, you cannot shrink it down to a point without breaking the loop or leaving the surface of the torus. Hence, a torus is not simply connected. This distinction between simply connected and non-simply connected spaces is fundamental in topology and has implications in various areas of mathematics and physics.The significance of simply connected spaces extends beyond theoretical mathematics. In complex analysis, for instance, the behavior of analytic functions is closely related to the topology of the underlying space. If a domain in the complex plane is simply connected, then every loop can be continuously contracted to a point, which allows for the application of powerful theorems such as Cauchy's integral theorem. This theorem states that if a function is analytic on a simply connected domain, the integral of that function over any closed curve within the domain is zero. This property greatly simplifies calculations and provides deep insights into the nature of analytic functions.Moreover, in the study of differential geometry, simply connected manifolds are essential when discussing curvature and topology. For example, the classification of surfaces reveals that the sphere is simply connected, while the projective plane is not. These characteristics influence the types of geometric structures that can exist on these surfaces and have implications in the broader context of general relativity and cosmology.Understanding the idea of simply connected spaces also aids in grasping more complex concepts such as homotopy and fundamental groups. The fundamental group is a mathematical structure that encapsulates information about the loops in a space, and for simply connected spaces, this group is trivial, meaning it contains only the identity element. This simplicity makes simply connected spaces easier to analyze and understand compared to their more complicated counterparts.In conclusion, the concept of simply connected is a cornerstone in the study of topology and has far-reaching implications across various fields of mathematics and science. By recognizing the characteristics and significance of simply connected spaces, we gain valuable insights into the structure and behavior of mathematical objects. Whether in the realm of pure mathematics or its applications in physics, the understanding of simply connected spaces serves as a gateway to deeper exploration and discovery.

在拓扑学这一迷人的数学分支中，简单连通的概念在理解不同空间的性质中扮演着至关重要的角色。一个拓扑空间被称为简单连通，如果它既是路径连通的，又没有“孔”。这意味着该空间中的任何循环都可以连续收缩到一个点，而不离开空间。为了说明这个想法，考虑一下放在平面上的橡皮筋。如果你用橡皮筋画一个环，你可以很容易地将其缩小到一个点，而无需将其从表面上抬起。因此，平面是简单连通的。另一方面，想象一个圆环，它的形状像一个甜甜圈。如果你在圆环的孔周围画一个循环，你无法在不打破循环或离开圆环表面的情况下将其缩小到一个点。因此，圆环不是简单连通的。这种简单连通和非简单连通空间之间的区别在拓扑学中是基础性的，并对数学和物理的各个领域产生了影响。简单连通空间的重要性超越了理论数学。在复分析中，例如，解析函数的行为与底层空间的拓扑密切相关。如果复平面中的一个域是简单连通的，那么任何循环都可以连续收缩到一个点，这允许应用强大的定理，如柯西积分定理。这个定理指出，如果一个函数在简单连通的域上是解析的，那么该函数在域内的任何闭合曲线上的积分都是零。这个属性大大简化了计算，并提供了对解析函数性质的深入见解。此外，在微分几何的研究中，简单连通流形在讨论曲率和拓扑时至关重要。例如，表面的分类揭示了球面是简单连通的，而射影平面则不是。这些特征影响着这些表面上可以存在的几何结构类型，并在广义相对论和宇宙学的更广泛背景下产生了影响。理解简单连通空间的概念还有助于掌握更复杂的概念，如同伦和基本群。基本群是一个数学结构，它封装了关于空间中循环的信息，对于简单连通空间来说，这个群是平凡的，意味着它仅包含单位元素。这种简单性使得简单连通空间比其更复杂的对应物更易于分析和理解。总之，简单连通的概念是拓扑学研究中的基石，并在数学和科学的各个领域具有深远的影响。通过认识简单连通空间的特征和重要性，我们获得了对数学对象的结构和行为的宝贵见解。无论是在纯数学领域还是其在物理学中的应用，理解简单连通空间都为更深层次的探索和发现提供了一个入口。