求高手英文翻译数学专业论文摘要！（望能手工翻译，数学专业名词要翻译正确，感激万分）

Question

连续性，紧性，致密性和完备性在数学分析实数理论，泛函分析和拓扑学中都有涉及。本文对以上四个性质在高等数学范畴内的三个领域进行详细的探究和讨论。
本文先就大学本科高等数学阶段所涉及到的三个领域内的四个性质进行了简单的梳理并予以适当的扩充。
首先，本文阐述了实数系的连续系定理，这包含六个方面的内容：确界定理，单调有界定理，闭区间套定理，有线覆盖定理，致密性定理和柯西收敛原理。并且在本章节的最后指出六个实数系连续性定理的等价性。
然后，进入泛函分析的范畴，首先讲诉了距离空间的基本概念以及收敛性，并在气候依次叙述了距离空间点集的相关概念。在本章的后两节，则详细阐述了距离空间的完备性和紧性。
第四章则设涉及拓扑学领域，从度量空间，拓扑空间和连续映射说起，依次提到紧性及几个紧性空间的关系，并给予出了详细的证明。并在本章节的最后部分点出度量空间的完备性，全有界性和紧性。
第五章是对前四张内容的总结，并进一步详细叙述在数学分析实数理论，泛函分析和拓扑学中，对连续性，紧性，致密性和完备性中进行横向比较和对其中关系的说明……

关键词：连续性，紧性，致密性，完备性

Answer 1

译文如下：

Continuity, compactness, density and completeness in the mathematical analysis theory of real number, functional analysis and topology have involved. In this paper, the above four properties in the higher mathematics field of three areas to explore and discuss the detailed.
This paper first university degree higher mathematics stage involved in three areas of the four properties of simple and the expansion of the comb appropriately.
First of all, the paper expounds the real number is in the continuous theorem, this includes six aspects of content: the reason that define a definition, drab principle, the closed interval, set of theorem, cable coverage theorem, density theorem and cauchy convergence theory. And at last, it points out that in this chapter six real continuation theorem of equivalence.
Then, into the category of functional analysis, first tell him about the basic concept of distance space and convergence, and in turn in climate described a set distance space related concepts. In this chapter, then after two detail the distance of the space completeness and compactness.
Chapter 4 involving topology domain, from a metric space, the topological space and continuous mapping in order to start, mentioned compactness and several compactness of the space, and to give out the relationship between the detailed proof. And in this chapter points the last part of the metric space the completeness, all the boundedness and compactness.
Chapter 5 of the content of the first four zhang is summarized, and further details in the mathematical analysis theory of real number, functional analysis and topology, the continuity, compactness, density and completeness of transverse comparison and on one of them to the relationship that......

Keywords: continuity, compactness, density, completeness

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