# Homework Help: Open set is a collection of regions

1. Dec 7, 2011

### mizunoami

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

Let U be a nonempty open set. The U is the union of a collection of regions.

2. The attempt at a solution

Let x be an element of U. There exists a region R such that x is in R and R is in U. So x is also in the union of a collection of regions Rx. Then U is a subset of Rx → This is the part I don't understand even though the class agrees upon it.

Also, since for all x, x is in R and is in U, so Rx is a subset of U. → I don't understand this part either!

Therefore U = Rx
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Dec 7, 2011

3. Dec 7, 2011

### HallsofIvy

What is your definition of "region"?

4. Dec 7, 2011

### mizunoami

If a,b \in C and a<b, then the set of points between a and b is the region ab.

Thanks!

5. Dec 7, 2011

### Dick

Every x in U is in a region contained in U. Doesn't that make the union of all of those regions U?

6. Dec 7, 2011

### mizunoami

Yeah. Then why is U a subset of the union of regions?

Since I'm showing U=union of regions, I want to show that U is a subset of the union of regions, and the union of regions is the subset of U.

Thanks a lot!

7. Dec 7, 2011

### Dick

Because every element of U is in one of the regions. So U is a subset of the union of the regions. Since every region is a subset of U, then the union of the regions is a subset of U.

8. Dec 7, 2011

### mizunoami

This TOTALLY makes my day. WOOHOO!

THANKS :)