Uncountably Many Copies of $\mathbb{R}$ Is the disjoint union of uncountably many copies of $\mathbb{R}$ metrizable? 
Edit: As suggested by the comments, I am referring to the disjoint union topology (I am asking whether the disjoint union of uncountably many copies of $\mathbb{R}$ with disjoint union topology is metrizable if $\mathbb{R}$ has the usual topology)
 A: Yes, for instance put a metric bounded by $1$ on $\mathbb{R}$, and then on your disjoint union put this metric on each copy of $\mathbb{R}$, and put distance $2$ on any two elements of different copies.
A: The answer depends on what you mean by "disjoint," and I think the distinction itself is even worth highlighting.
I suspect that [this][1]https://en.wikipedia.org/wiki/Long_line_(topology) is what you have in mind. This is a set-theoretic disjoint union, but not a topological one. This is not metrizable.
$\mathbb{R^2}$ is a disjoint union of uncountably many copies of the real line, and it certainly has a metric.
If on the third and final hand you mean a metric space homeomorphic to the topological disjoint union (this is not, bu the way, homeomorphic to either of the other spaces) then you can consider a metric space product of the real line with a discrete uncountable space.
In other words, you could say points on the same line have their usual distance, and for points on different lines, just use any fixed positive number.
