Closure

In topology, the closure of a set $A$ is the intersection of all closed sets containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . Equivalently, the closure of $A$ is the union of $A$ and all limit points of $A$

A closure operator is an abstract (category theory) form of the topological notion of closure which can be applied to any set $A$. It is a function $cl$ from $A$ to the power set of $A$ satisfying the following conditions:

• $A\in cl(A)$ (augmentation)
• If $B\subset A$ then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cl(B)\subset cl(A)} (monotonicity)
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cl(A)\in cl(A)} (idempotence)

In the category of topological spaces, this operator is isomorphic to the standard topological one.