Focusing on the construction of Borel sets in n dimensional Euclidean space， we propose several results of measure theory features for detailed discussion. Utilizing the good structure of set class Al which consists of those n dimensional leftopen and rightclosed intervals such that left end point is mi/2l（where mi is integer and l is positive integer） and length of each side is 1/2l， combined with partition of real line， inequality techniques and topological techniques，we first prove that Al is a countably infinite partition of n dimensional Euclidean space for each positive integer l， and as l gets larger，Al gets finer， and for each point in each open subset of the n dimensional Euclidean space，the member in Al who contained the point must be contained by the open subset when l is sufficiently large. Then， based on the previous results we prove that every open subset in n dimensional Euclidean space can be expressed as the union of at most countably infinite n dimensional leftopen and rightclosed intervals by way of contradiction. Last， arming with this theorem， we introduce some generators for the Borel algebra of n dimensional Euclidean Space.