Open subset

Clopen sets and non-open and/or non-closed sets

A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset and a closed subset. Such subsets are known as clopen sets. Explicitly, a subset ${\displaystyle S}$ of a topological space ${\displaystyle (X,\tau )}$ is called clopen if both ${\displaystyle S}$ and its complement ${\displaystyle X\setminus S}$ are open subsets of ${\displaystyle (X,\tau )}$; or equivalently, if ${\displaystyle S\in \tau }$ and ${\displaystyle X\setminus S\in \tau .}$