Closed set

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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) \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).
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