Assume that $2^{\omega\_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the *support* of $y$, and let $X$={$x\in Y:0<|\operatorname{supp}(x)|\le\omega\_1$}; $|X|=\mathfrak{c}^{\omega\_1}=(2^{\omega\_1})^{\omega\_1}=\mathfrak{c}$. 


Does X have $G\_\delta$ diagonal?



Does X have any diagonal properties?
For a space $X$ to have a $G\_\delta$ diagonal, it needs (although this is not enough) to have countable pseudo-character (i.e. every point must be a $G\_\delta$). This is clearly not the case for your space.


