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Closed set

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Examples

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 $Γ$ 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
$\bigcap_{\gamma \in \Gamma} (0,a_{\gamma}]\cup [b_{\gamma},1).$ .
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
$C = \big\{ f \in C[a,b] \mid \min_{x \in [a,b]} f(x) \ge 0 \}$
and
$D = \big\{ f \in C[a,b] \mid \min_{x \in [a,b]} f(x) \le 0 \}$
are closed (the sets $C$ and $D$ are the closure of the sets $A$ and $B$ respectively).