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.

$\blacksquare$

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: $2 \ \text{(a)}$