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!