Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic
Theorem
Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
- $\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Let $\tau_d$ denote the metric topology for $d$.
Let $\tau_\delta$ denote the metric topology for $\delta$.
Then $\struct {\Z_{>0}, \tau_d}$ and $\struct {\Z_{>0}, \tau_\delta}$ are homeomorphic.
Proof
From Topology induced by Usual Metric on Positive Integers is Discrete:
- $\struct {\Z_{>0}, \tau_d}$ is a discrete space.
From Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete:
- $\struct {\Z_{>0}, \tau_\delta}$ is a discrete space.
Let $I_{\Z_{>0} }$ be the identity mapping from $\Z_{>0}$ to itself.
From Mapping from Discrete Space is Continuous:
- $I_{\Z_{>0} }: \struct {\Z_{>0}, \tau_d} \to \struct {\Z_{>0}, \tau_\delta}$ is continuous
and:
- $I_{\Z_{>0} }: \struct {\Z_{>0}, \tau_\delta} \to \struct {\Z_{>0}, \tau_d}$ is continuous.
Hence the result by definition of homeomorphic.
$\blacksquare$
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: $10$