Open sets in R^2: Are open sets contained in their closures?

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In summary, the discussion is about open sets in R^2 and whether if A is contained in the closure of B, then it is also contained in B. One person believes it is intuitively true, while another person gives a counterexample to show it is not always true.
  • #1
ehrenfest
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[SOLVED] open sets in R^2

Homework Statement


If A, B are open in R^2, and A is contained in the closure(B), then is it true that A is contained in B?

Homework Equations


The Attempt at a Solution


It seems intuitively true.

If a is in A, then there is a sequence b_n in B that converges to a by definition of closure. But I don't know what to do with that?
 
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  • #2
It doesn't seem intuitively true to me. Take B=R^2-{(0,0)}. Give me a counterexample.
 
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  • #3
Dick said:
It doesn't seem intuitively true to me. Take B=R^2-{(0,0)}. Give me a counterexample.

any open set containing the origin
 
  • #4
Yep.
 

1. What are open sets in R^2?

Open sets in R^2 are subsets of the two-dimensional Euclidean space that do not include their boundaries. In other words, every point within an open set must have a certain distance from the boundary of the set.

2. How are open sets defined mathematically?

In mathematical terms, a set S in R^2 is considered open if for every point p in S, there exists a positive distance r such that all points q within r units of p are also in S.

3. What is the difference between an open set and a closed set?

The main difference between an open set and a closed set is that open sets do not include their boundaries, while closed sets do. Another way to think about it is that every point in an open set has a "neighborhood" around it that is also contained within the set, whereas in a closed set, some points may not have a neighborhood within the set boundaries.

4. Can open sets have infinite elements?

Yes, open sets in R^2 can have an infinite number of elements. For example, a circle in R^2 can be considered an open set, and a circle contains an infinite number of points.

5. How are open sets used in mathematical analysis?

Open sets are an important concept in mathematical analysis, particularly in the study of limits, continuity, and differentiation. They also play a role in topology, where they are used to define continuous functions and topological spaces. In general, open sets help mathematicians to define and analyze mathematical objects in a more precise and rigorous manner.

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