Countable Complement Topology is Expansion of Finite Complement Topology
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Theorem
Let $T = \struct {S, \tau}$ be the countable complement topology on an infinite set $S$.
Let $T' = \struct {S, \tau'}$ be the finite complement topology on the same infinite set $S$.
Then $\tau$ is an expansion of $\tau'$.
Proof
Let $U \in \tau', U \ne \O$.
Then $\relcomp S U$ is finite by definition of finite complement topology.
Then by definition of countable complement topology, we have that $U \in \tau$.
So $\tau' \subseteq \tau$ and hence the result by definition of expansion.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $1$