Proving Properties of Open Sets in Real Numbers

[saadsarfraz]

Homework Statement

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

(a) Show that, in the above dedefinition, the numbers a, b may be taken
as rational; that is, if x $$\in$$ U, there is an open interval (c, d) where
x $$\in$$ (c, d) $$\subseteq$$ U and where c, d $$\in$$ Q.

(b) Show that any open set U is a union of (possibly infinitely many)
intervals (a, b) where a, b $$\in$$ Q.

(c) How many open subsets of R exist?

Homework Equations

The Attempt at a Solution

i don't have much idea, the idiot prof hasnt even covered most of the stuff in class.

 

[CompuChip]

For (a), let m := (b - a) / 2 be the midpoint of the interval. Do there exist rational numbers c and d such that a < c < m < d < b?

For (b), here's a hint:
$$U = \bigcup_{x \in U} x$$.

 

[saadsarfraz]

Hi, thank you for the hints but I am still stuck on part c. any ideas?

 

[CompuChip]

I take it that means that you did a and b.

I haven't given c much though myself. You could start by counting how many open intervals there are, for which it suffices to count intervals of the form (c, d) with c and d rational. Then how many unions can you take?