Proving the Intersection of Closed Sets is Closed | Homework Solution

Thread starter Pjennings
Start date Nov 15, 2009
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Closed Sets

In summary, a closed set is a set that contains all of its limit points. When the intersection of two sets is closed, it means that all limit points of the intersection are also contained in the intersection. To prove that the intersection of closed sets is closed, one must show that all limit points of the intersection are also contained in the intersection. An example of the intersection of closed sets being closed is [0,1] and [0,2] resulting in [0,1]. Proving that the intersection of closed sets is closed is significant in real analysis and topology, as it allows us to make conclusions about the properties of sets and their limits.

Nov 15, 2009

Pjennings

Homework Statement

Show that the intersection of two closed sets is closed.

Homework Equations

The Attempt at a Solution

Let X and Y be closed sets i.e. X and Y are equal to their closure X_ and Y_. Then X[tex]\cap[/tex]Y is equal to X_[tex]\cap[/tex]Y_.

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Nov 15, 2009

Dick

Science Advisor

Homework Helper

And how would that show X intersect Y is closed?

1. What is the definition of a closed set?

A closed set is a set that contains all of its limit points. A limit point is a point that can be approached arbitrarily close by points in the set.

2. What does it mean for the intersection of two sets to be closed?

When the intersection of two sets is closed, it means that all limit points of the intersection are also contained in the intersection.

3. How do you prove that the intersection of closed sets is closed?

To prove that the intersection of closed sets is closed, you must show that all limit points of the intersection are also contained in the intersection. This can be done by assuming a point is a limit point of the intersection and then showing that it must also be contained in the intersection.

4. Can you provide an example of the intersection of closed sets being closed?

Yes, an example would be the intersection of the closed sets [0,1] and [0,2], which results in the closed set [0,1]. All limit points of the intersection [0,1] are also contained in [0,1], making it closed.

5. What is the significance of proving that the intersection of closed sets is closed?

Proving that the intersection of closed sets is closed is important in real analysis and topology. It allows us to show that certain operations on closed sets, such as taking intersections, will result in another closed set. This can help us make conclusions about the properties of sets and their limits.