Direct sum of metrizable spaces. I managed to prove that an arbitrary direct sum of metrizable spaces is again metrizable. However, I used the theorem that says that a hausdorff regular space is metrizable if and only if there existd a $\sigma$-locally finite base. I was planning to show my proof for some friends, but they never saw the proof of this equivalence, therefore I was trying to create a simpler proof.
I am also trying to show that every direct sum of completely metrizable spaces is again completely metrizable and I am having no success.

It seems the following.
Let $\{X_\alpha\}$ be a family of (completely) metrizable spaces. Then for each $\alpha$ there exists a (complete on $X_\alpha$) compatible metric $d_\alpha$. Then 
$$d(x,y)=\cases{
\min\{d_\alpha(x,y),1\},\text{ if }x,y\in X_\alpha\mbox{ for some }\alpha\\ 
2,\mbox{ otherwise }}
$$
is a (complete) compatible metric on the direct sum of the family $\{X_\alpha\}$.
