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Clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of ''open'' and ''closed'' are antonyms. But their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is itself also open, making both sets both open ''and'' closed, and therefore clopen.
==Examples==
In any topological space ''X'', the empty set and the whole space ''X'' are both clopen.〔 (regarding the real numbers and the empty set in R)〕〔 (regarding topological spaces)〕
Now consider the space ''X'' which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on ''X'' is inherited as the subspace topology from the ordinary topology on the real line R. In ''X'', the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.
As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, and the set ''A'' of all positive rational numbers whose square is bigger than 2. Using the fact that is not in Q, one can show quite easily that ''A'' is a clopen subset of Q. (Note also that ''A'' is ''not'' a clopen subset of the real line R; it is neither open nor closed in R.)
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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