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:

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

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