User:IssaRice/Adherent point and limit point

Let $(X, d)$ be a metric space, let $E$ be a subset of $X$ , and let $x_{0} \in X$ be a point.

Adherent point

there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of points in $E$ which converges to $x_{0}$
for every radius $r > 0$ the ball $B (x_{0}, r)$ has nonempty intersection with $E$
$x_{0}$ is an interior point of $E$ or is a boundary point of $E$
for every open set $U$ such that $x \in U$ one has $U \cap E \neq \emptyset$

Limit point

for every open set $U$ such that $x \in U$ there is some $y \in U \cap E$ such that $y \neq x$
for every open set $U$ such that $x \in U$ , the set $U \cap E$ has infinitely many points
there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of distinct points in $E$ (i.e. $x_{n} \in E$ for all $n \geq 1$ and $x_{n} \neq x_{m}$ for all $n \neq m$ ) which converges to $x_{0}$
there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of points in $E$ , none of which are equal to $x_{0}$ , which converges to $x_{0}$

Limit point of a sequence

$x_{0}$ is a limit point of $(x_{n})_{n = 1}^{\infty}$ iff for every $ε > 0$ and every $N \geq 1$ there exists $n \geq N$ such that $d (x_{n}, x) \leq ε$

$x_{0}$ is a limit point of $(x_{n})_{n = 1}^{\infty}$ iff for every $N \geq 1$ , $x_{0}$ is an adherent point of ${a_{n} : n \geq N}$