What is the cardinality of $X$? Let $X=\{x\in D^\mathfrak{c}:
0<|\{\xi<\mathfrak{c}:x(\xi)=1\}|\le\omega_1\}$, where $D=\{0,1\}$.
What is the cardinality of $X$? I think it is $\mathfrak c$, however I'm not sure. Also I don't know how to prove it. Thanks for your help.

$|X|=\mathfrak{c}^{\omega_1}=(2^\omega)^{\omega_1}=2^{\omega_1}$. This can be $\mathfrak{c}$, if for instance, $2^\omega=2^{\omega_1}=\omega_2$, which is consistent with $\mathsf{ZFC}$, but under $\mathsf{CH}$ we have $2^{\omega_1}=2^{\mathfrak{c}}>\mathfrak{c}$.
