question about $Y$-homogeneous spaces. A subspace $Y$ of space $X$ is $h$-dense in $X$, if $Y$ is dense in $X$ and, for each $x\in X$, there exists a homeomorphism $h$ of $X$ onto itself such that $h(x)\in Y$. in this case we say that $X$ is $Y$-homogeneous.
Let $Y$ is a $G_\delta$-dense subspace of $Y$-homogeneous space $X$ and $U$ be an open subset of $X$ and $x\in \overline{U}$, is it true that $x\in Y$ ?
See: Topological groups and C-embeddings, Theorem 1.5.
How do you use the definition of $Y$-homogeneous to show that $x\in Y$?
In general, Is there a logical relation between homogeneous spaces and $Y$-homogeneous spaces?
thanks for the advise.
 A: It isn’t necessarily true that $x\in Y$. However, $X$ is $Y$-homogeneous, so there is a homeomorphism $h$ of $X$ onto itself such that $h(x)\in Y$, and we replace $x$ and $U$ by $h(x)$ and $h[U]$. Arhangel’skii’s argument in the paper then shows that there is a $G_\delta$-set $P$ in $X$ such that $h(x)\in P\subseteq h[\operatorname{cl}U]$, and it follows immediately that $h^{-1}[P]$ is a $G_\delta$-set $P$ in $X$ such that $x\in h^{-1}[P]\subseteq\operatorname{cl}[U]$.

Clearly a homogeneous space $X$ is $Y$-homogeneous for every dense $Y\subseteq X$. I don’t 
Added: The converse is false. Let $X=\{0,1\}^\omega$ and $Y=\{0,1\}^{\omega_1}$, each with the product topology, and let $Z=X\sqcup Y$. If $D$ is a dense subset of $Z$, then $Z\cap X$ and $Z\cap Y$ are dense in $X$ and $Y$, respectively. $X$ and $Y$ are homogeneous, so for any $x\in X$ and $y\in Y$ there are homeomorphisms $h_X$ and $h_Y$ of $X$ and $Y$ onto themselves such that $h_X(x)\in D\cap X$ and $h_Y(y)\in D\cap Y$, and clearly $h_X\cup h_Y$ is a homeomorphism of $Z$ onto itself taking $x$ and $y$ to points of $D$. However, there is clearly no homeomorphism of $Z$ onto itself taking $x$ to $y$. Thus, $Z$ is $D$-homogeneous for each dense $D\subseteq Z$, but $Z$ is not homogeneous.
