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Paul
Paul
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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$={$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$G_\delta$ 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 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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Paul
Paul
  • 654
  • 4
  • 15

Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D=\{ 0,1 \}$$D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)=\{\xi<\mathfrak{c}:y(\xi)=1\}$$\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?

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?

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?

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Paul
Paul
  • 654
  • 4
  • 15

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 any diagonal properties?