|
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. ==Definition== The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.〔.〕 A modern definition is as follows. An open cover of a topological space ''X'' is a family of open sets whose union contains ''X''. The ''ply'' of a cover is the smallest number ''n'' (if it exists) such that each point of the space belongs to at most ''n'' sets in the cover. A refinement of a cover ''C'' is another cover, each of whose sets is a subset of a set in ''C''; its ply may be smaller than, or possibly larger than, the ply of ''C''. The covering dimension of a topological space ''X'' is defined to be the minimum value of ''n'', such that every finite open cover ''C'' of ''X'' has a refinement with ply ''n'' + 1 or below. If no such minimal ''n'' exists, the space is said to be of infinite covering dimension. As a special case, a topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement. 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「Lebesgue covering dimension」の詳細全文を読む スポンサード リンク
|