Closed set

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

Examples

1. Let $X=(0,1)$ with the usual topology induced by the Euclidean distance. Open sets are then of the form $\cup _{\gamma \in \Gamma }(a_{\gamma },b_{\gamma })$ where 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 0\leq a_{\gamma}\leq b_{\gamma} \leq 1} 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 \Gamma} is an arbitrary index set (if 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=b} then define 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,b)=\emptyset} ). Then closed sets by definition are of the form 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 \cap_{\gamma \in \Gamma} (0,a_{\gamma}]\cup [b_{\gamma},1)} .

2. As a more interesting example, consider the function space 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[a,b]} consisting of all real valued continuous functions on the interval [a,b] (a<b) endowed with a topology induced by the distance 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(f,g)=\mathop{\max}_{x \in [a,b]}|f(x)-g(x)|} . In this topology, 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=\{ g \in C[a,b] \mid \mathop{\min}_{x \in [a,b]}g(x) > 0 \}}

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=\{ g \in C[a,b] \mid \mathop{\min}_{x \in [a,b]}g(x) < 0\}}

are open sets while 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=\{ g \in C[a,b] \mid \mathop{\min}_{x \in [a,b]}g(x) \geq 0\}=C[a,b]-B}

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=\{ g \in C[a,b] \mid \mathop{\min}_{x \in [a,b]}g(x) \leq 0\}=C[a,b]-A}

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 closures 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} ).

See also

Topology

Analysis

Open set

Compact set