Category:Definitions/Limit Points in Metric Spaces
This category contains definitions related to Limit Points in Metric Spaces.
Related results can be found in Category:Limit Points in Metric Spaces.
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 $\alpha \in S$.
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 $\struct{S, \tau}$.
Pages in category "Definitions/Limit Points in Metric Spaces"
The following 4 pages are in this category, out of 4 total.