# How many open sets of R do exist?

I was reading a set theory book and they claim that there are as many open sets in $$\mathbb R$$ as real numbers (usual topology).

I tried using the base of intervals with irrational extremes but found nothing.

Any hint is welcome.

• Every open subset of $\mathbb{R}$ can be written as a union of open intervals with rational endpoints. – saulspatz Mar 26 at 2:17
• That's not true. $\mathbb{R}$ and $\mathbb{R} -\sqrt{2}$ are different open sets but contain the same set of rationals. – David Lui Mar 26 at 3:06
• @DavidLui Thanks. :-( – user759562 Mar 26 at 3:13
• @DavidLui They do indeed contain the same set of rationals, but that doesn't stop them from being expressed as unions of open intervals with rational endpoints. For example, $\mathbb R \backslash \{\sqrt{2}\}$ is the union of the open intervals $(-n, a_n)$ and $(b_n, n)$ where $a_n$ and $b_n$ are increasing and decreasing sequences, respectively, of rationals with limit $\sqrt{2}$. – Robert Israel Mar 26 at 3:49
• Sine the other person deleted their post, I was responding to another comment saying that an open set was determined by the rationals that it contained. – David Lui Mar 26 at 4:03

As |{ ($$\infty$$,r ) : r in R }| = c = |R|,
c$$^{|N|}$$ = c.