G.M. Reed developed a construction technique which associates a Moore space $\mathcal M(X)$ to each regular first-countable space $X$ such that $\mathcal M(X)$ is separable (respectively, locally separable, CCC, or DCCC) if and only if $X$ has the corresponding property.
The construction is as follows. Let $X$ be a regular first-countable space. For each $x\in X$, denote by $\{U_n(x)\}$ a sequence of open sets in $X$ which forms a local base at $x$ such that for each $n$, $\overline{U_{n+1}(x)} \subset U_n(x)$. Now, for each $m\in \mathbb{N}^+$, let $A_m=\{(n_1, n_2, \cdots, n_m): n_1=1 \text{ and for } 1 \le i \le m, n_i \in \mathbb{N}^+\}$. Let $A=\bigcup_{1\le m < \omega}A_m$. For each $a= (n_1, n_2, \cdots, n_m) \in A$, denote by $S_a$ a unique copy of $X$ such that all copies are pairwise disjoint, and for each $x\in X$, denote by $(x_{n_1}, x_{n_2}, \cdots, x_{n_m})$ the element of $S_a$ which is identified with $x$. Let $\mathcal M(X)= \bigcup \{S_a: a\in A\}$ and define a development for $\mathcal M(X)$ as follows: For each $j \in \mathbb{N}^+$, $a= (n_1, n_2, \cdots, n_m) \in A$ and $p=(y_{n_1}, y_{n_2}, \cdots, y_{n_m}) \in S_a$, let \begin{align*} &G_j(p)=\{p\} \cup \{(x_{n_1}, x_{n_2}, \cdots, x_{n_m}, x_{k_1}, x_{k_2}, \cdots, x_{k_c}): x\in X, c\in \mathbb{N}^+, \\ & \text{ for } 1\le i\le c, k_i \ge j \text{ and }x\in U_{k_1+j}(y)\}. \end{align*} It follows that $\mathcal B=\{G_j(p): p\in \mathcal M(X) \text{ and } j\in \mathbb{N}^+\}$ is a basis for a topology on $\mathcal M(X)$ and that $\{\mathcal G_n\}$, where for each $n$, $\mathcal G_n=\{G_j(p): p\in \mathcal M(X) \text{ and } j \ge n\}$, is a development for the Moore space $\mathcal M(X)$.
A cellular family is a family of pairwise disjoint non-empty open sets. The cellularity of a space $X$ ($c(X)$) is defined as the supremum of the cardinalities of the cellular families in $X$.
It eas pointed out that the cellularity of $\mathcal M(X)$ is countable iff the cellularity of $X$ is countable. My question is this:
Is $c(\mathcal M(X)) = c(X)$ for any first countable regular space?
