Determine whether set is closed, open or neither.

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In summary, a closed set contains all of its limit points and includes its boundary points, while an open set does not. To determine if a set is closed, check if its complement is open. To determine if a set is open, check if all of its points have a neighborhood contained within the set. An example of a closed set is the set of all integers, while an example of an open set is the set of all real numbers between 0 and 1. A set can also be both open and closed, known as a clopen set.
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Homework Statement


D1 = {(x,y) : x^2 + y^2 < 3, x+2y = 2}
D2={(x,y) : x^2 + y^2 > 2}
D3={(x,y) : x + 2y = 2}

Homework Equations

The Attempt at a Solution


D1 is neither, D2 is open and D3 is closed, am I right or wrong?
 
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Right!
 
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1. What is the difference between a closed and open set?

A closed set is a set that contains all of its limit points, while an open set is a set that does not contain all of its limit points. In other words, a closed set includes its boundary points, while an open set does not.

2. How do you determine if a set is closed?

A set is closed if its complement is open. In other words, if all the points not in the set have a neighborhood that is not contained in the set, then the set is closed.

3. How do you determine if a set is open?

A set is open if all of its points have a neighborhood that is completely contained in the set. In other words, if all the points in the set are interior points, then the set is open.

4. What is an example of a closed set?

An example of a closed set is the set of all integers. This set contains all of its limit points (since integers are closed under addition and subtraction), and its complement (the set of all non-integer numbers) is open.

5. Can a set be both open and closed?

Yes, a set can be both open and closed. This is known as a clopen set. An example of a clopen set is the set of all real numbers between 0 and 1, including both 0 and 1.

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