Let $\{X\}_{\alpha\in\mathscr{A}}$ be a collection of metrizable spaces and consider their disjoint union $$X:=\bigsqcup_{\alpha\in\mathscr{A}}X_{\alpha}.$$ Show that $X$ is metrizable.  Not sure where to begin. I figured some employment of a theorem is need; such as, the Urysohn Metrization Theorem, but I don't see how that could be used. Any help is appreciated.
I have encountered this question while tring to figure out why every sequential space $X$, is a quotient spaces of a metric space. If I understand correctly, given a sequential space, every sequence (including it's limit) can be identified with the space $Y=\{0\} \cup \{\frac1{n+1}|n \in N\}$. So The space $X$ can viewed as a quotient space of $\bigoplus_{(x_n) \in C} \{(x_n) \} \times Y$ where $C$ is the set of all converges sequences in $X$. My question is, How can I show that this space is metrizable?  Thank you! Shir
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