同胚

The circle and the square are not homeomorphic because they have different topological properties.
圆形和正方形不是同胚的，因为它们有不同的拓扑性质。
The concept of homeomorphism is fundamental in topology, allowing us to classify shapes based on their connectivity.
同胚的概念在拓扑学中至关重要，它允许我们根据形状的连接性对形状进行分类。
A continuous deformation that transforms one shape into another without tearing or gluing is an example of a homeomorphism.
没有撕裂或粘合将一个形状连续变形为另一个形状的过程就是一个同胚的例子。
In proving that two spaces are homeomorphic, it's often necessary to find a specific function that acts as a homeomorphism between them.
在证明两个空间是同胚的过程中，通常需要找到一个特定的函数作为它们之间的同胚函数。
The torus (doughnut shape) and the coffee cup are homeomorphic because they both have one hole.
饼干形状的环面（torus）与咖啡杯是同胚的，因为它们都有一个洞。
Two surfaces are considered homeomorphic if there exists a continuous bijection with a continuous inverse between them.
两个表面被认为是同胚的，如果它们之间存在一个连续的一一对应关系，并且这个对应关系的逆也连续。
Homeomorphisms preserve topological properties such as connectedness and compactness, but not necessarily the size or shape of the objects.
同胚保持拓扑性质如连通性和紧致性，但不一定保持对象的大小或形状。
The study of homeomorphisms helps mathematicians understand the underlying structure of spaces and how they can be transformed.
对同胚的研究帮助数学家理解空间的基本结构以及它们如何被转换。
Topologists often use homeomorphisms to prove that two spaces are equivalent for certain purposes, despite looking very different.
拓扑学家经常使用同胚来证明，在某些目的下，尽管看起来非常不同，两个空间是等价的。
The concept of homeomorphism is used in various fields, including computer graphics, where it helps in creating realistic deformations of objects.
同胚的概念在各个领域都有应用，包括计算机图形学，在这里它有助于创建物体的真实变形。