英语翻译实数的连续性是实数集最基本也是最重要的性质之一,是实数集区别于有理数集的重要特征.实数集关于极限运算是封闭的,因

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英语翻译
实数的连续性是实数集最基本也是最重要的性质之一,是实数集区别于有理数集的重要特征.实数集关于极限运算是封闭的,因此将极限理论建立在实数集之上,极限理论便有了牢固的基础.拓扑学是近代发展起来的一个研究连续性现象的数学分支,主要研究拓扑空间在拓扑变换下的不变性质和不变量.从拓扑的观点来看实数的连续性定理,我们将站在更高的角度来理解和掌握实数的连续性的本质.
本文首先介绍了实数的连续性的发展历史,给出实数的连续性的六个等价定理,接着讲述点集拓扑学的一些基本概念以及相关定理,最后使用这些基本概念和定理重新阐示实数的连续性定理,以达到深入理解的目的.
翻译以上,

The continuity of real mumber is one of the most fundamental and important properties in the set of real numbers,and also is an important character in comprarison with the set of all rational numbers.The set of real numbers is closed in limit operation,so limit theory who is built on the set of real numbers has a solid foundation.Topology,as a branch of mathematica of researching continuity developed in modern time,is to research the invariable property and invariants of topological space under the topological transformation.From the topological point to research the continuity theorem of real number,the essence of continuity would be well understood and known at a higher perspective.
In this paper,the development history of continuity of real mumber was firsty introduced,and Six Equivalence Theorems about continuity were given.Then some basic concepts and theorems of general topology were told.At last,with these basic concepts and theorems,continuity of real mumber was mainly explained so that continuity theorem of real mumber was deeply understood.
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数域的完备性大一数学书上有句话，有理数域不是封闭的（即有理数的数串的极限可能不再是有理数），而实数域对极限运算是封闭的（什么是纯实数集..纯实数集和实数集的交集是空集? {实数集}{全体实数}所表示的集合是实数集R吗?为什么〔实数集〕与〔实数〕是一样的含义么,有什么区别实数集与有理数集的区别与联系有理数与实数的区别 R*表示的是正实数集还是实数集里排除0的集合? 实数.有理数的定义证明f(x)=x^3+1是实数集到实数集的双射 “单调有界收敛准则是实数集R的一个重要属性”, 复数集是实数集的扩充,因此复数在保留实数的一些性质的同时,也使得实数的一些性质在复数集上不能成立. 实数指数幂的运算性质