Components of Integer Reciprocal Space with Zero are Single Points

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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 the components of $A$ are singletons.


From Integer Reciprocal Space with Zero is Totally Separated, $A$ is totally separated.

From Totally Separated Space is Totally Disconnected, $A$ is totally disconnected.

The result follows by definition of totally disconnected.