Is this set countable? (intervals in $\mathbb{R}$)

Question

A set consists of intervals in $\mathbb{R}$. They have no intersection, their starting point and end point can be any numbers(not limited to rational number), and the union all intervals should cover all real numbers. For example, the first interval is $[0,1)$ then the next interval can be $[1,6)$, but not $[0.5,2)$. How many such intervals? Is it countable infinity?

Is a single point an interval? If not, then each "interval" contains a rational so... — lulu, Mar 17 '21 at 11:24

score 1 · Accepted Answer · 2021-03-17T11:57:49.773

1

The same rational number cannot be in two of these intervals. That gives us an injection from the set of intervals to $\mathbb Q$. Thus there are at most the cardinality of the rationals. But there are only countably many rationals.

Note: the axiom of choice may rear its head here.

edited Mar 17 '21 at 11:57

answered Mar 17 '21 at 11:38

Is this set countable? (intervals in $\mathbb{R}$)

1 Answers1