Definition:Closed Set

Topology
A subset $$X$$ of a topological space $$Y$$ is closed if its complement relative to $Y$ is in the topology of $$Y$$.

Complex Analysis
A specific application of this concept is found in the field of complex analysis.

Let $$S \subseteq \mathbb{C}$$ be a subset of the set of complex numbers.

Then $$S$$ is closed (in $$\mathbb{C}$$) iff its complement $$\mathbb{C} - S$$ is open in $$\mathbb{C}$$.

Alternative definition
A subset $$S \subseteq \mathbb{C}$$ is closed iff it contains all of its limit points.