- #1
proplaya201
- 14
- 0
Homework Statement
how do you show a set is open in R^n?
Demonstrating Openness in R^n Sets
- Thread starter proplaya201
- Start date
-
- Tags
- Sets
Click For Summary
Homework Help Overview
The discussion revolves around demonstrating that a set is open in R^n, particularly within the context of metric spaces and topology as outlined in Rudin's "Principles of Mathematical Analysis." Participants are exploring definitions and approaches related to open sets and interior points.
Discussion Character
- Conceptual clarification, Assumption checking, Exploratory
Approaches and Questions Raised
- Participants discuss various methods to show a set is open, including using the definition of an open set, demonstrating the existence of a neighborhood around points in the set, and considering the complement of the set. Questions arise about specific definitions and the application of these concepts.
Discussion Status
The conversation is ongoing, with participants seeking clarification on definitions and methods. Some guidance has been offered regarding starting points for proofs, but there is no explicit consensus on a single approach yet.
Contextual Notes
Participants are referencing specific definitions of open sets and interior points, indicating a focus on foundational concepts in topology and metric spaces. There is a request for more specificity in approaches, highlighting the need for clarity in the context of different mathematical frameworks.
how do you show a set is open in R^n?
- #2
CompuChip
Science Advisor
Homework Helper
- 4,305
- 49
Depends on which course you are taking.
In analysis you could show that for every point in the set there is some small ball completely contained inside the set.
In topology you could work from the definition of open set, use a basis, show that the complement is closed, or even use some more elaborate theorem.
So please be a little more specific
In analysis you could show that for every point in the set there is some small ball completely contained inside the set.
In topology you could work from the definition of open set, use a basis, show that the complement is closed, or even use some more elaborate theorem.
So please be a little more specific
- #3
proplaya201
- 14
- 0
im reading rudin's book: principles in mathematical analysis, ad we are talking about metric spaces, ie topology. so can you expand on you second approach to the problem please?
- #4
HallsofIvy
Science Advisor
Homework Helper
- 42,895
- 983
The same way you prove almost anything: use the definition of open set. What is the definition of open set you are using?
- #5
proplaya201
- 14
- 0
a set is open if every point in the set is an interior point. now i know that but i am having difficulty proving it.
(every point being an interior point that is)
(every point being an interior point that is)
- #6
CompuChip
Science Advisor
Homework Helper
- 4,305
- 49
So the proof would start like: "Let x be any point in the set ..." and then shows that x satisfies the definition of an interior point.
What is the definition of an interior point?
What is the definition of an interior point?
- #7
proplaya201
- 14
- 0
then there exists a neighborhood of x such that neighborhood of x is contained in the set
Similar threads
- · Replies 1 ·
- · Replies 8 ·
- · Replies 5 ·
- · Replies 1 ·
- · Replies 13 ·