Category:Equivalence of Definitions of Limit Point in Metric Space

From ProofWiki
Jump to navigation Jump to search

This category contains pages concerning Equivalence of Definitions of Limit Point in Metric Space:


Let $M = \struct {S, d}$ be a metric space.

Let $\tau$ be the topology induced by the metric $d$.

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.


The following definitions of the concept of limit point in metric space are equivalent:

Definition 1

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

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

that is:

$\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$

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


Definition 2

$\alpha$ is a limit point of $A$ if and only if there is a sequence $\sequence{\alpha_n}$ in $A \setminus \set \alpha$ such that $\sequence{\alpha_n}$ converges to $\alpha$, that is, $\alpha$ is a limit of the sequence $\sequence{\alpha_n}$ in $S$.


Definition 3

$\alpha$ is a limit point of $A$ if and only if $\alpha$ is a limit point in the topological space $\struct{S, \tau}$.