Definition:Odd-Even Topology
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Definition
Let $\Z_{>0}$ denote the set of strictly positive integers:
- $\Z_{>0} = \set {x \in \Z: x > 0}$
Let $\PP$ be the partition on $\Z_{>0}$ defined as:
- $\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$
That is:
- $\PP = \set {\set {1, 2}, \set {3, 4}, \set {5, 6}, \ldots}$
Then the topology whose basis is $\PP$ is called the odd-even topology.
Also see
- Results about the odd-even topology can be found here.
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: $3$