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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