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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. A subset of a topological space ''X'' is a connected set if it is a connected space when viewed as a subspace of ''X''. An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced by two-dimensional Euclidean space. ==Formal definition== A topological space ''X'' is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, ''X'' is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topological space ''X'' the following conditions are equivalent: #''X'' is connected. #''X'' cannot be divided into two disjoint nonempty closed sets. #The only subsets of ''X'' which are both open and closed (clopen sets) are ''X'' and the empty set. #The only subsets of ''X'' with empty boundary are ''X'' and the empty set. #''X'' cannot be written as the union of two nonempty separated sets (sets for which each is disjoint from the other's closure). #All continuous functions from ''X'' to are constant, where is the two-point space endowed with the discrete topology. 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「Connected space」の詳細全文を読む スポンサード リンク
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