Set theory

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 de definition, 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 in finitely 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.
 

CompuChip

Science Advisor
Homework Helper
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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:
[tex]U = \bigcup_{x \in U} x[/tex].
 
Hi, thank you for the hints but im still stuck on part c. any ideas?
 

CompuChip

Science Advisor
Homework Helper
4,284
47
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?
 

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