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 any diagonal properties?