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.

Proof
We have:
 * Integer Reciprocal Space with Zero is Totally Separated
 * Totally Separated Space is Totally Disconnected
 * Totally Disconnected and Locally Connected Space is Discrete

We also have:
 * Topological Space is Discrete iff All Points are Isolated
 * Zero is Limit Point of Integer Reciprocal Space

From definition of limit points:
 * $0$ is not an isolated point of $A$

Hence integer reciprocal space with zero is not the discrete space, and the result follows.