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proplaya201
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Homework Statement
how do you show a set is open in R^n?
Demonstrating Openness in R^n Sets
- Thread starter proplaya201
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In summary, to show that a set is open in R^n, it is necessary to prove that every point in the set is an interior point. This can be done by using the definition of an interior point and showing that for any point x in the set, there exists a neighborhood of x that is completely contained in the set. This can also be approached by using the definition of an open set and proving that the complement of the set is closed. In analysis, this can be done by showing that for every point in the set, there is a small ball completely contained in the set. In topology, a basis can be used or more elaborate theorems can be applied.
how do you show a set is open in R^n?
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CompuChip
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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
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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
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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
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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
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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
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then there exists a neighborhood of x such that neighborhood of x is contained in the set
1. What is the concept of openness in R^n sets?
The concept of openness in R^n sets refers to a set of points in n-dimensional space where every point in the set has a neighborhood that is also contained within the set. In other words, every point in the set is surrounded by other points in the same set. This concept is important in topology and analysis, as it helps define continuity and convergence of functions.
2. How do you demonstrate openness in R^n sets?
To demonstrate openness in R^n sets, one must show that for any point in the set, there exists a neighborhood of that point that is also contained within the set. This can be proven through various methods, such as using the definition of openness or using the concept of open balls.
3. What is the difference between an open set and a closed set in R^n?
An open set in R^n is a set that contains all of its limit points, while a closed set is a set that contains all of its boundary points. In other words, an open set does not include its boundary points, while a closed set does. Another way to think about it is that an open set is "open" in the sense that it has no boundaries, while a closed set is "closed" in the sense that it includes its boundaries.
4. Can a set be both open and closed in R^n?
Yes, a set can be both open and closed in R^n. This type of set is known as a clopen set. An example of a clopen set in R^2 is the entire plane, which includes all of its boundary points (the x and y axes) and has no boundaries at the same time.
5. How does demonstrating openness in R^n sets relate to real-world applications?
Demonstrating openness in R^n sets has many real-world applications, particularly in fields such as physics, engineering, and computer science. For example, in physics, the concept of open sets is used to describe the behavior of continuous systems, while in computer science, it is used to define algorithms for finding optimal solutions in complex spaces. Additionally, the concept of openness is also used in data analysis and machine learning to identify patterns and relationships in large datasets.
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