Odd-Even Topology is not Countably Compact
Jump to navigation
Jump to search
Theorem
Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is not countably compact.
Proof
By definition, the odd-even topology is a partition topology.
Let $\PP$ be the partition which is the basis for $T$:
- $\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$
Then $\PP$ is a countable open cover of $S$ which has no finite subcover.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $6$. Odd-Even Topology: $3$