|
In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers } generated by the family of all arithmetic progressions. It is a special case of the profinite topology on a group. This particular topological space was introduced by where it was used to prove the infinitude of primes. == Construction == An arithmetic progression associated to two whole numbers ''a'' and ''k'', with is the set of integers : To give the set Z a topology means to say which subsets of Z are open in a manner that satisfies the following axioms: # The union of open sets is an open set. # The finite intersection of open sets is an open set. # Z and the empty set ∅ are open sets. The family of all arithmetic progressions does not satisfy these axioms: the union of arithmetic progressions need not be an arithmetic progression itself, e.g., } is not an arithmetic progression. So the evenly spaced integer topology is defined to be the topology ''generated by'' the family of arithmetic progressions. This is the coarsest topology that includes as open subsets the family of all arithmetic progressions: that is, arithmetic progressions are a subbase for the topology. Since the intersection of any finite collection of arithmetic progressions is again an arithmetic progression, the family of arithmetic progressions is a base for the topology, meaning that every open set is a union of arithmetic progressions.〔〔For general background on bases and subbases, see .〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「evenly spaced integer topology」の詳細全文を読む スポンサード リンク
|