# Is each Lindelof closed $\bar G_\delta$-set of a Tychonoff space functionally closed?

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.

• If $F$ is a Lindelof $G_\delta$-set then $F=f^{-1}(0)$ for some function $f$ of Baire class $1$. – Ramiro de la Vega Nov 10 '17 at 0:13
• @RamirodelaVega any reference to your statement above? – Idonknow Jan 6 at 1:18
• @Idonknow This follows from Propositions 2 and 4(b) in Extending Baire-one functions on topological spaces, O.F.K. Kalenda, J. Spurny, Topology and its Applications 149 (2005) 195–216. – Ramiro de la Vega Jan 15 at 15:58