Right Order Topology on Strictly Positive Integers is not Metrizable
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Theorem
Let $\Z_{>0}$ be the set of strictly positive integers.
Let $T = \struct {\Z_{>0}, \tau}$ denote the right order space on $\Z_{>0}$.
Then $T = \struct {\Z_{>0}, \tau}$ is not a metrizable space.
Proof
Let $m, n \in \Z_{>0}$ such that $m < n$.
Let $O_m$ and $O_n$ be arbitrary non-empty open sets of $T$.
Then:
- $O_m \cap O_n = O_m$
As $O_m$ and $O_n$ are arbitrary, it follows that there exist no $O_m$ and $O_n$ in $\tau$ such that $O_m \cap O_n = \O$.
Hence $T$ is not Hausdorff.
The result follows from Metrizable Space is Hausdorff.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Exercise $1$