How do I prove that there exists a bijection from power set of $D$ to $C'\subseteq C.$

Question

I was proving the Moore's Plane $(X,\mathscr T)$ is not normal. Using the theorem. Let $C=\{(x,y)\in X:y=0\}$ be a closed and relatively discrete subset of $X$. Let $D=\{(x,y)\in X:x\in \mathbb Q \text{ and } y\in \mathbb Q\}$ is a dense subset of $X$. How do I prove that there exists a bijection from power set of $D$ to $C'\subseteq C.$

Answer 1

The power set of $D$ has size continuum ($2^{\aleph_0}$) and so does $C \simeq \mathbb{R}$ so they're even bijectively related, they have the same size.

Answer 2

The set $D$ is countable, and so we can think of it as the set $\mathbb{N}$ of all natural numbers. Any subset of $\mathbb{N}$ can be associated uniquely with a sequence of $0$s and $1$s; Given $Y\subset \mathbb{N}$, define $f_Y$:

$$f_Y(x)=\begin{cases} 0, & x\notin Y,\\ 1, & x \in Y. \end{cases}$$

It is easy to check that $Y = Z$ if and only if $f_Y = f_Z$.

The function that takes a subset $Y$ of $\mathbb{N}$ and sends it to $(0.f_Y, 0)\in C$ satisfies your request.