Topology: An Inquiry-Based Approach

Let $X$ be a metric space and $A$ a subset of $X\text{.}$

Boundary points and limit points don’t need to be in the set $A\text{,}$ whereas an isolated point of $A$ must be in $A\text{.}$ In $A=(0,1)\cup \{2\}$ as a subset of $=(\mathbb{R},d_E)\text{,}$ $0$ is a boundary point but not an isolated point while $2$ is a boundary point but not a limit point. Also, $0.5$ is a limit point but neither a boundary or isolated point. With $A$ as subset of $\mathbb{R}$ with the discrete metric, every point of $A$ is an isolated point but no point in $\mathbb{R}$ is a boundary point or a limit point of $A\text{.}$ So even though every boundary point is either a limit point or an isolated point, the three concepts are different.