Closed sets

Open sets appear directly in the definition of a topological space. It next seems that closed sets are needed. Suppose $ X$ is a topological space. A subset $ C \subset X$ is defined to be a closed set if and only if $ X \setminus C$ is an open set. Thus, the complement of any open set is closed, and the complement of any closed set is open. Any closed interval, such as $ [0,1]$, is a closed set because its complement, $ (-\infty,0) \cup (1,\infty)$, is an open set. For another example, $ (0,1)$ is an open set; therefore, $ {\mathbb{R}}
\setminus (0,1) = (-\infty,0] \cup [1,\infty)$ is a closed set. The use of ``$ ($'' may seem wrong in the last expression, but ``$ [$'' cannot be used because $ -\infty$ and $ \infty$ do not belong to $ {\mathbb{R}}$. Thus, the use of ``$ ($'' is just a notational quirk.

Are all subsets of $ X$ either closed or open? Although it appears that open sets and closed sets are opposites in some sense, the answer is no. For $ X = {\mathbb{R}}$, the interval $ [0,2
\pi)$ is neither open nor closed (consider its complement: $ [2 \pi,\infty)$ is closed, and $ (-\infty,0)$ is open). Note that for any topological space, $ X$ and $ \emptyset$ are both open and closed!

Steven M LaValle 2012-04-20