# Closed set

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In mathematics, a set ${\displaystyle A\subset X}$, where ${\displaystyle (X,O)}$ is some topological space, is said to be closed if ${\displaystyle X-A=\{x\in X\mid x\notin A\}}$, the complement of ${\displaystyle A}$ in ${\displaystyle X}$, 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
${\displaystyle \bigcup _{\gamma \in \Gamma }(a_{\gamma },b_{\gamma })}$
where ${\displaystyle 0\leq a_{\gamma }\leq b_{\gamma }\leq 1}$ and ${\displaystyle \Gamma }$ is an arbitrary index set (if ${\displaystyle a=b}$ then the open interval (a, b) is defined to be the empty set). The definition now implies that closed sets are of the form
${\displaystyle \bigcap _{\gamma \in \Gamma }(0,a_{\gamma }]\cup [b_{\gamma },1).}$.
2. As a more interesting example, consider the function space ${\displaystyle C[a,b]}$ (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
${\displaystyle \|f\|=\max _{x\in [a,b]}|f(x)|.}$
In this topology, the sets
${\displaystyle A={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)>0\}}$
and
${\displaystyle B={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)<0\}}$
are open sets while the sets
${\displaystyle C={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)\geq 0\}}$
and
${\displaystyle D={\big \{}f\in C[a,b]\mid \min _{x\in [a,b]}f(x)\leq 0\}}$
are closed (the sets ${\displaystyle C}$ and ${\displaystyle D}$ are the closure of the sets ${\displaystyle A}$ and ${\displaystyle B}$ respectively).