Definition:Limit Point/Normed Vector Space/Set

Definition

Let $M = \struct {X, \norm {\,\cdot\,}}$ be a normed vector space.

Let $Y \subseteq X$ be a subset of $X$.

Let $\alpha \in X$.

Then $\alpha$ is a limit point of $Y$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$:

$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$

that is:

$\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$

Note that $\alpha$ does not have to be an element of $A$ to be a limit point.

(Informally speaking, $\alpha$ is a limit point of $Y$ if there are points in $Y$ that are different from $\alpha$ but arbitrarily close to it.)