Open set is a collection of regions

[mizunoami]

Homework Statement

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 R_x. Then U is a subset of R^x → 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 R_x is a subset of U. → I don't understand this part either!

Therefore U = R_x

 

[mizunoami]

Can someone please help? A new proof is good too. Thanks!

 

[HallsofIvy]

What is your definition of "region"?

 

[mizunoami]

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

Thanks!

 

[Dick]

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

 

[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!

 

[Dick]

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!

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.

 

[mizunoami]

Because every element of U is in one of the regions. So U is a subset of the union of the regions.

This TOTALLY makes my day. WOOHOO!

THANKS :)