Adherent, Accumulation, and Isolated Points in Metric Spaces

# Adherent, Accumulation, and Isolated Points in Metric Spaces

Recall from the Adherent Points of Subsets in Euclidean Space page that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{x} \in \mathbb{R}^n$ is an adherent point of $S$ if there exists an $\mathbf{s} \in S$ such that $\mathbf{s} \in B(\mathbf{x}, r)$ for all positive real numbers $r > 0$, i.e., every ball centered at $\mathbf{x}$ contains at least one element from $S$.

Also recall from the Accumulation Points of Subsets in Euclidean Space page that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{x} \in \mathbb{R}^n$ is an accumulation point of $S$ if there exists an $\mathbf{s} \in S \setminus \{ \mathbf{x} \}$ such that $\mathbf{s} \in B(\mathbf{x}, r)$ for all positive real numbers $r > 0$, i.e., every ball centered at $\mathbf{x}$ contains at least one element from $S$ different from $\mathbf{x}$.

Lastly, recall from the Isolated Points of Subsets in Euclidean Space page that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{x} \in \mathbb{R}^n$ is an isolated point of $S$ if there exists a positive real number $r_0 > 0$ such that $B(\mathbf{x}, r_0)$ contains no points from $S$ other than $\mathbf{x}$.

We will now define all of these points in terms of general metric spaces. The definitions below are analogous to the ones above with the only difference being the change from the Euclidean metric to any metric.

 Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $x \in M$ is an Adherent Point of $S$ if there exists an $s \in S$ such that $s \in B(x, r) = \{ y \in M : d(x, y) < r \}$ for all positive real numbers $r > 0$, i.e., every ball centered at $x$ contains at least one element from $S$.

Equivalently, $x \in M$ is an adherent point of $S$ if for all $r > 0$, $B(x, r) \cap S \neq \emptyset$.

 Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $x \in M$ is an Accumulation Point (or Limit Point) of $S$ if there exists an $s \in S \setminus \{ x \}$ such that $s \in B(x, r) = \{ y \in M : d(x, y) < r \}$ for all positive real numbers $r > 0$, i.e., every ball cenetered at $x$ contains at least one element from $S$ different from $x$.

Equivalently, $x \in M$ is an accumulation point of $S$ if for all $r > 0$, $B(x, r) \cap (S \setminus \{ x \}) \neq \emptyset$.

 Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $x \in S$ is an Isolated Point of $S$ if there exists a positive real number $r_0 > 0$ such that $B(x, r) = \{ y \in M : d(x, y) < r_0 \}$ contains no points from $S$ other than $x$.

Equivalently, $x \in S$ is an isolated point of $S$ if there exists an $r_0 > 0$ such that $B(x, r_0) \cap (S \setminus \{ x \}) = \emptyset$.