Integer Reciprocal Space with Zero is not Locally Connected

Theorem
Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
 * $A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$

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

Then $A$ is not locally connected.