Open set is a collection of regions

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Homework Help Overview

The discussion revolves around the concept of open sets in topology, specifically focusing on the characterization of an open set U as a union of regions. Participants are exploring the implications of this definition and the relationships between U and the regions that comprise it.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to understand the relationship between an open set and the regions it contains. There are questions regarding the definitions of "region" and the reasoning behind why U is a subset of the union of these regions. Some express confusion about the implications of their reasoning and seek clarification on the proof structure.

Discussion Status

The discussion is active, with participants questioning definitions and the logical flow of their arguments. Some guidance has been provided regarding the relationships between U and the regions, but there is still uncertainty about the proof and the definitions involved.

Contextual Notes

Participants are working within the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is an emphasis on understanding the definitions and relationships rather than providing a complete proof.

mizunoami
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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 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
 
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Can someone please help? A new proof is good too. Thanks!
 
What is your definition of "region"?
 
If a,b \in C and a<b, then the set of points between a and b is the region ab.

Thanks!
 
Every x in U is in a region contained in U. Doesn't that make the union of all of those regions U?
 
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!
 
mizunoami said:
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.
 
Dick said:
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 :)
 

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