7
$\begingroup$

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.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ If $F$ is a Lindelof $G_\delta$-set then $F=f^{-1}(0)$ for some function $f$ of Baire class $1$. $\endgroup$
    – Ramiro de la Vega
    Nov 10, 2017 at 0:13
  • $\begingroup$ @RamirodelaVega any reference to your statement above? $\endgroup$
    – Idonknow
    Jan 6, 2019 at 1:18
  • $\begingroup$ @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. $\endgroup$
    – Ramiro de la Vega
    Jan 15, 2019 at 15:58

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.