Not every Open Set is F-Sigma Set

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $V$ be an open set of $T$.

Then it is not necessarily the case that $V$ is an $F_\sigma$ set of $T$.

Proof

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.

Let $U$ be an open set of $T$.

From Closed Set of Uncountable Finite Complement Topology is not $F_\sigma$:

$U$ is not an $F_\sigma$ set of $T$.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction