# Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval

## Theorem

Let $A \subseteq \R$ be the set of all points on $\R$ defined as:

- $A := \left\{{\dfrac 1 n : n \in \Z_{>0}}\right\}$

Let $\left({A, \tau_d}\right)$ be the integer reciprocal space under the usual (Euclidean) topology.

Let $B$ be the uncountable set:

- $B := A \cup \left[{2 \,.\,.\, 3}\right]$

where $\left[{2 \,.\,.\, 3}\right]$ 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:

- $I := \left({- a \,.\,.\, b}\right) \subseteq U$

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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 32: \ 1 \ \text{(b)}$