2 ms·

Cardinality hasn’t much to do with it since there are uncountable sets which you may cover like that.

• Koshkin 5y ago
  Sure, but for countable subsets (such as the rationals) it is easy to show.
  • l33t2328 5y ago
    Sure. Explaining what’s different about the reals is the hard part
    • Koshkin 5y ago
      How so? Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length. (Unless the person you are trying to explain this to starts raising questions about the meaning of 'interval' or 'length', in which case the meaning of the original question becomes just as uncertain in the first place.)
      • l33t2328 5y ago
        > Even a real interval of a finite length cannot be covered by any set of intervals of a smaller total length. You’ve only restated the problem without saying _why_ covering the reals is different.
        • Koshkin 5y ago
          Because the 'intervals' in question are already 'real'?