Open Set of Uncountable Finite Complement Topology is not F-Sigma
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.
Let $U \in \tau$ be an open set of $T$.
Then $U$ is not an $F_\sigma$ set.
Proof
Let $U$ be an open set of $T$.
As $S$ is uncountable, then so is $U$.
By the definition of a finite complement topology, all closed sets of $T$ are finite.
From Countable Union of Countable Sets is Countable, $U$ can not be expressed as the union of a countable number of closed sets.
So by definition $U$ is not an $F_\sigma$ set.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $19$. Finite Complement Topology on an Uncountable Space: $3$