Closed set

In mathematics, a set 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 \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 $0\leq a_{\gamma }\leq b_{\gamma }\leq 1$ and $\Gamma$ is an arbitrary index set (if $a=b$ then define $(a,b)=\emptyset$). Then closed sets by definition are of the form $\cap _{\gamma \in \Gamma }(0,a_{\gamma }]\cup [b_{\gamma },1)$.

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

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

and

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

are open sets while the sets

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

and

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

are closed (the sets $C$ 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