Editing Clopen set (section)

{{short description|Subset which is both open and closed}}
{{Distinguish|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 set|open]] and [[closed set|closed]].  That this is possible may seem counterintuitive, as the common meanings of {{em|open}} and {{em|closed}} are antonyms, but their mathematical definitions are not [[mutually exclusive]].  A set is closed if its [[Complement (set theory)|complement]] is open, which leaves the possibility of an open set whose complement is also open, making both sets both open {{em|and}} closed, and therefore clopen. As described by topologist [[James Munkres]], unlike a [[door]], "a set can be open, or closed, or both, or neither!"{{sfn|Munkres|2000|p=91}} emphasizing that the meaning of "open"/"closed" for {{em|doors}} is unrelated to their meaning for {{em|sets}} (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "[[door space]]s" their name.