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.

$\blacksquare$

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: $1$