Sequence of Imaginary Reciprocals
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Theorem
Consider the subset $S$ of the complex plane defined as:
- $S := \set {\dfrac i n : n \in \Z_{>0} }$
That is:
- $S := \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 4, \ldots}$
where $i$ is the imaginary unit.
Then $S$ has the following properties:
Boundedness
The set $S$ is bounded in $\C$.
Limit Points
The set $S$ has exactly one limit point, and that is $z = 0$.
Closedness
The set $S$ is not closed.
Boundary Points
Every point of $S$, along with the point $z = 0$, is a boundary point of $S$.
Interior
No point of $S$ is an interior point.
Openness
$S$ is not an open set.
Connectedness
$S$ is not connected.
Not an Open Region
$S$ is not an open region.
Closure
The closure of $S$ is the set:
- $\set {0, i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$
Complement
The complement of $S$ in $\C$ is the set:
- $\C \setminus \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$
Countability
The set $S$ is countably infinite.
Not Compact
The set $S$ is not compact.
Closure is Compact
The closure $S^-$ of the set $S$ is compact.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Point Sets: $45$