# Integer Reciprocal Space is Topological Space

## Theorem

Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.

Let $A \subseteq \R$ be the set of all points on $\R$ defined as:

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

Then the integer reciprocal space $\struct {A, \tau_d}$ is a topological space.

## Proof

We have that $A \subseteq \R$.

By definition, $\struct {A, \tau_d}$ is a subspace of $\struct {\R, \tau_d}$.

Hence the result from Topological Subspace is Topological Space.

$\blacksquare$