Final answer:
A set A is closed if and only if it includes all of its boundary points, which is shown through a two-part proof involving both the definition of closed sets and the property of containing limit points.
Step-by-step explanation:
In mathematics, specifically in topology, the closure of a set A in a topological space is the smallest closed set containing A. A set is defined as closed if and only if it contains all its boundary points. Boundary points are those that can be approached both from within the set and from its exterior. To show that a set A is closed if and only if it contains all of its boundary points, we can use two directions of proof:
- If A is closed, then by definition, it must contain all its boundary points since a closed set includes its boundary.
- Conversely, if A contains all its boundary points, then there is no limit point of A that lies outside of A, and hence it must be closed because it encompasses all points that are limit points of sequences within A.
Therefore, a set A is closed if and only if it contains all of its boundary points, completing the proof.