Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval
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Theorem
Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
- $A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Let $B$ be the uncountable set:
- $B := A \cup \closedint 2 3$
where $\closedint 2 3$ is a closed interval of $\R$.
$2$ and $3$ are to all intents arbitrary, but convenient.
Then $0$ is not a condensation point of $B$ in $\R$.
Proof
Let $U$ be an open set of $\R$ which contains $0$.
From Open Sets in Real Number Line, there exists an open interval $I$ of the form:
- $I := \openint {-a} b \subseteq U$
From Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval, there is a countably infinite number of points of $B$ in $U$.
However, when $b < 2$ there is not an uncountable number of points of $B$ in $I$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $1 \ \text{(b)}$