Closed set

In mathematics, a set $A\subset X$, where $(X,O)$ is some topological space, is said to be closed if $X-A=\{x\in X\mid x\notin A\}$, the complement of $A$ in $X$, is an open set. The empty set and the set X itself are always closed sets. The finite union and arbitrary intersection of closed sets are again closed.

Examples

1. Let X be the open interval (0, 1) with the usual topology induced by the Euclidean distance. Open sets are then of the form

   $\bigcup _{\gamma \in \Gamma }(a_{\gamma },b_{\gamma })$

   where $0\leq a_{\gamma }\leq b_{\gamma }\leq 1$ and $\Gamma$ is an arbitrary index set (if $a=b$ then the open interval (a, b) is defined to be the empty set). The definition now implies that closed sets are of the form

   $\bigcap _{\gamma \in \Gamma }(0,a_{\gamma }]\cup [b_{\gamma },1).$.

2. As a more interesting example, consider the function space $C[a,b]$ (with a < b). This space consists of all real-valued continuous functions on the closed interval [a, b] and is endowed with the topology induced by the norm

   $\|f\|=\max _{x\in [a,b]}|f(x)|.$

   In this topology, the sets

   $A={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)>0\}$

   and

   $B={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)<0\}$

   are open sets while the sets

   $C={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)\geq 0\}$

   and

   $D={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)\leq 0\}$

   are closed (the sets $C$ and $D$ are the closure of the sets $A$ and $B$ respectively).