Topology: An Inquiry-Based Approach

In addition to limit points, we also defined boundary points in metric spaces. Recall that a boundary point of a set $A$ in a metric space $X$ could be considered to be any point in $\bar{A}\cap \overline{X\setminus A}\text{.}$ We make the same definition in a topological space.

As with metric spaces, the boundary points of a set $A$ are those points that are “between” $A$ and its complement.

Just as with metric spaces, we can characterize the closed sets as the sets that contain their boundary.

The proof of Theorem 13.11 is left to Exercise 10.