Sunday, August 21, 2016


Clopen set

graph with several clopen sets. Each of the three large pieces (i.e. components) is a clopen set, as is the union of any two or all three.
Not to be confused with Half-open interval.
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 openand 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 also open, making both sets both open and closed, and therefore clopen.


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