Is $[0,1]$ a countable disjoint union of closed sets? Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?

This is just a consequence of the fact that a Cantor like set is uncountable.
In our case one creates the Cantor set by intersecting compact sets formed by subtracting from the [0,1] the union of first N intervals without their boundary points.
Since the intersection is uncountable it contains a point which is not the end point of any intervals. That is, there is a point in [0,1] which does not belong to the sum of closed intervals.
