How can i prove it?
[Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G_{\delta}-dense $ in $X$.
How can i prove it?
[Gillman and Jerison] If a dense subspace $Y$ of a Tychonoff space $X$ is $C-embedded$ in X, then $Y$ is $ G_{\delta}-dense $ in $X$.
$\newcommand{\cl}{\operatorname{cl}}$Suppose that $G$ is a non-empty $G_\delta$ in $X$ such that $G\cap Y=\varnothing$. Then there are open sets $U_n$ for $n\in\Bbb N$ such that $G=\bigcap_{n\in\Bbb N}U_n$ and $U_n\supseteq U_{n+1}$ for each $n\in\Bbb N$. Fix $x\in G$. Since $X$ is Tikhonov, for each $n\in\Bbb N$ there is a continuous function $f_n:X\to[0,1]$ such that $f_n(x)=1$ and $f_n(y)=0$ for all $y\in X\setminus U_n$. Now define
$f:Y\to\Bbb R:y\mapsto\sum_{n\in\Bbb N}f_n(y)\;;$
each $y\in Y$ belongs to only finitely many of the sets $U_n$, so $f$ is well-defined. Does $f$ have a continuous extension to $X$?