Topology: An Inquiry-Based Approach

There is a connection between limit points and closed sets. The open set $(1,2)$ in $(\mathbb{R},d_E)$ does not contain all of its limit points or any of its boundary points, while the closed set $[1,2]$ contains all of its boundary and limit points. This is an important attribute of closed sets. Recall that for a limit point $x$ of a subset $A$ of a metric space $X\text{,}$ every neighborhood of $x$ contains a point in $A$ different from $x\text{.}$ We can make the neighborhoods as small as we like so, in a sense, the limit points of $A$ that are not in $A$ are the points in $X$ that are arbitrarily close to the set $A\text{.}$ We denote the set of limit points of $A$ as $A^{\prime }\text{,}$ and the limit points of a set can tell us if the set is closed.