- 86

- 1

**1. The problem statement, all variables and given/known data**

A subset U [tex]\subseteq[/tex] R is called open if, for every x [tex]\in[/tex] U, there is an open interval (a, b) where x [tex]\in[/tex] (a, b) [tex]\subseteq[/tex] U.

(a) Show that, in the above dedefinition, the numbers a, b may be taken

as rational; that is, if x [tex]\in[/tex] U, there is an open interval (c, d) where

x [tex]\in[/tex] (c, d) [tex]\subseteq[/tex] U and where c, d [tex]\in[/tex] Q.

(b) Show that any open set U is a union of (possibly infinitely many)

intervals (a, b) where a, b [tex]\in[/tex] Q.

(c) How many open subsets of R exist?

**2. Relevant equations**

**3. The attempt at a solution**

i dont have much idea, the idiot prof hasnt even covered most of the stuff in class.