# Does X have any diagonal properties?

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?

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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.