Definition:Limit Point/Complex Analysis
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Definition
Let $S \subseteq \C$ be a subset of the set of complex numbers.
Let $z_0 \in \C$.
Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$.
Then $z_0$ is a limit point of $S$ if and only if every deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$:
- $\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$
that is:
- $\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$
Note that $z_0$ does not itself have to be an element of $S$ to be a limit point, although it may well be.
Informally, there are points in $S$ which are arbitrarily close to it.
Also known as
A limit point is also known as:
However, note that an accumulation point is also seen with a subtly different definition from that of a limit point, so be careful.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $2.$