Abstract: The notions of general important degree of an attribute and attributes subset are introduced, and thereupon a kind of general Euclidean distance and a general neighborhood relation are set up. And then a real rough set theory based on the general neighborhood relation is proposed. The theory partitions the universe into tolerant modules under general neighborhood relationship the give the lower and upper approximation of the set without discretizing the data. Furthermore, the definition and greedy arithmetic of attribute reduction in real rough set theoryare proposed, and the quality of reduction results is analyzed. Finally, an example is given to show the validity and effectiveness of the theory and method of this paper.