A subset $F$ if a topological space $X$ is called *functionally closed* if $F=f^{-1}(0)$ for some continuous map $f:X\to[0,1]$.

It is clear that each functionally closed set $F$ in $X$ is a closed $G_\delta$-set in $X$ and moreover, $F$ is a $\bar G_\delta$-set.

A subset $F$ of a topological space $X$ will be called a *$\bar G_\delta$-set* if $F=\bigcap_{n\in\omega}U_n=\bigcap_{n\in\omega}\bar U_n$ for some open sets $U_n$, $n\in \omega$, in $X$.

It can be proved that a $\sigma$-compact subset $F$ of a Tychonoff space $X$ is functionally closed if and only if $F$ is a $\bar G_\delta$-set in $X$.

Can this characterization be generalized to Lindelof or cosmic spaces?

**Problem.** Is every Lindelof (cosmic) $\bar G_\delta$-set $F$ in a Tychonoff space $X$ functionally closed?

We recall that a space is *cosmic* if it has a countable network of the topology. It is well-known that each cosmic space is Lindelof.

Extending Baire-one functions on topological spaces, O.F.K. Kalenda, J. Spurny, Topology and its Applications 149 (2005) 195–216. $\endgroup$ – Ramiro de la Vega Jan 15 at 15:58