Closed set: Difference between revisions
imported>Hendra I. Nurdin (→See also: Added link to open set) |
imported>Hendra I. Nurdin (→Examples: Corrected example 1 to include the empty set (''a''<''b''-->'a''<=''b'') and fixed the closed sets) |
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== Examples == | == Examples == | ||
1. Let <math>X=(0,1)</math> with the usual topology induced by the Euclidean distance. Open sets are then of the form <math>\cup_{\gamma \in \Gamma} (a_{\gamma},b_{\gamma})</math> where <math>0\leq a_{\gamma} | 1. Let <math>X=(0,1)</math> with the usual topology induced by the Euclidean distance. Open sets are then of the form <math>\cup_{\gamma \in \Gamma} (a_{\gamma},b_{\gamma})</math> where <math>0\leq a_{\gamma}\leq b_{\gamma} \leq 1</math> and <math>\Gamma</math> is an arbitrary index set (if <math>a=b</math> then define <math>(a,b)=\emptyset</math>). Then closed sets by definition are of the form <math>\cap_{\gamma \in \Gamma} (0,a_{\gamma}]\cup [b_{\gamma},1)</math>. | ||
2. As a more interesting example, consider the function space <math>C[a,b]</math> consisting of all real valued [[continuous function|continuous functions]] on the interval [a,b] (a<b) endowed with a topology induced by the distance <math>d(f,g)=\mathop{\max}_{x \in [a,b]}|f(x)-g(x)|</math>. In this topology, the sets | 2. As a more interesting example, consider the function space <math>C[a,b]</math> consisting of all real valued [[continuous function|continuous functions]] on the interval [a,b] (a<b) endowed with a topology induced by the distance <math>d(f,g)=\mathop{\max}_{x \in [a,b]}|f(x)-g(x)|</math>. In this topology, the sets | ||
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are closed (the sets <math>C</math> and <math>D</math> are, respectively, the [[closures|closure]] of the sets <math>A</math> and <math>B</math>). | are closed (the sets <math>C</math> and <math>D</math> are, respectively, the [[closures|closure]] of the sets <math>A</math> and <math>B</math>). | ||
== See also == | == See also == | ||
Revision as of 16:15, 3 September 2007
In mathematics, a set , where is some topological space, is said to be closed if , the complement of in , is an open set
Examples
1. Let with the usual topology induced by the Euclidean distance. Open sets are then of the form where and is an arbitrary index set (if then define ). Then closed sets by definition are of the form .
![{\displaystyle \cap _{\gamma \in \Gamma }(0,a_{\gamma }]\cup [b_{\gamma },1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2716d7e1b0effe943b868992eaecf85c3410731)
2. As a more interesting example, consider the function space consisting of all real valued continuous functions on the interval [a,b] (a<b) endowed with a topology induced by the distance . In this topology, the sets
![{\displaystyle C[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1625217aad8c105c50c975599e45192b2bfbec)
![{\displaystyle d(f,g)=\mathop {\max } _{x\in [a,b]}|f(x)-g(x)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88230b50a590e64fc034ee3ddd100a4ed6c6d7fa)
and
are open sets while the sets
and
are closed (the sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} are, respectively, the closure of the sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} ).