|
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. == Definition == A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let τ1 and τ2 be two topologies on a set ''X'' such that τ1 is contained in τ2: :. That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1. 〔There are some authors, especially analysts, who use the terms ''weak'' and ''strong'' with opposite meaning (Munkres, p. 78).〕 If additionally : we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1.〔 The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on ''X''. 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「comparison of topologies」の詳細全文を読む スポンサード リンク
|