Real Analysis/Open and Closed Sets

Terminology The open ball in a metric space with radius centered at a, is denoted . Formally Definition Let be a metric space. We say a set is open if for every such that . We say a set is closed if is open. A closed set can also be defined as a set which contains all of its limit points. This property follows directly from the previous definition, as if , there exists a radius of which contains no point of , so p cannot be a limit point of , and all limit points of are contained within .