Metrization problem: $Y$ is a $T_0$-space and a union of closed, nbd-finite, metrizable subspaces. I've found the following problem in James Dugundji's Topology (Chap. IX, section 9, problem 4).

Prove: If a $T_0$-space $Y$ is the union of a nbd-finite family of closed metrizable subspaces, then $Y$ is also metrizable.

A family of subsets si said to be nbd-finite if every point of the space has a nbd intersecting at most finitely many elements of the family.
I've been trying for a while with no success but I believe I need to use the following theorem (IX, 9.2 in the cited text)

Theorem (K. Morita) $Y$ is metrizable if and only if $Y$ is a $T_0$-space, and there exists a sequence $\{\mathfrak{F}_n\mid n\in Z^+\}$ of nbd-finite closed coverings with the property: for each $y\in Y$ and nbd $W(y)$ there is an $n$ such that $\text{St}(y,\mathfrak{F}_n)\subseteq W$.

In this context, $\text{St}(y,\mathfrak{F}_n)=\{x\in Y\mid \exists F\in \mathfrak{F}_n,\; (y\in F\land x\in F)\}$, or equivalently, $\text{St}(y,\mathfrak{F}_n)$ is the union of all those elements of $\mathfrak{F}_n$ that contain $y$.
My attempts consist of trying to play around with the coverings of each metrizable subspace and trying to create a sequence of coverings for the entire space with the desired characteristics. I'm looking for any hints or solutions.
 A: Suppose $\mathfrak{F}$ is a neighborhood-finite cover of $Y$ by closed metrizable subsets.  For each $C\in\mathfrak{F}$, let $\mathfrak{F}_{C,n}$ be a neighborhood-finite cover of $C$ by closed sets of diameter at most $1/n$ (with respect to some chosen metric on $C$).  Let $\mathfrak{F}_n=\bigcup_{C\in\mathfrak{F}}\mathfrak{F}_{C,n}$.  Then $\mathfrak{F}_n$ is neighborhood-finite: each $y\in Y$ has a neighborhood that intersects only finitely many $C_1,\dots,C_m\in\mathfrak{F}$, and then $y$ has a neighborhood which intersects only finitely many elements of $\mathfrak{F}_{C_i,n}$ for $i=1,\dots,m$.  Finally, given $y\in Y$ and a neighborhood $W$ of $y$, there are only finitely many $C_1,\dots,C_m\in\mathfrak{F}$ that contain $y$.  There is then some $n$ such that $W$ contains the ball of radius $1/n$ around $n$ in each of $C_1,\dots,C_m$ (with respect to our chosen metrics on them).  Then every element of $\mathfrak{F}_n$ which contains $y$ is contained in $W$, since it is in some $\mathfrak{F}_{C_i,n}$ and thus has diameter at most $1/n$ with respect to the metric of $C_i$.  Thus by Theorem IX.9.2, $Y$ is metrizable.
(By the way, the $T_0$ assumption is unnecessary, since the existence of a closed metrizable set containing each point immediately implies the space is $T_1$.)
