Closed set

In mathematics, a set $$A \subset X$$, where $$(X,O)$$ is some topological space, is said to be closed if its complement in $$X$$, the set $$X-A=\{x \in X \mid x \notin A\}$$, is open. 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
  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 $$0 < a_{\gamma} < b_{\gamma} < 1,$$ and $$\Gamma$$ is an arbitrary index 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).  