Prove that countable intersections of closed subset of R^d are closed

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In summary, a closed subset of R^d contains all its boundary points, and a countable intersection is a collection of subsets being intersected together with a countably infinite number of subsets. It is important to prove that countable intersections of closed subsets of R^d are closed because it is a fundamental property of topological spaces and allows for generalizations. This can be proven by showing that the limit point of the intersection of countably infinite closed subsets is also within the intersection. This property holds true in any topological space, although a countable basis is necessary for it to hold.
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doggie_Walkes
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Prove that countable intersections of closed subset of R^d are closed
 
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Try proving that countable unions of open subsets of R^d are open.
 
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If the intersection is not empty (what happens if it is?), consider a convergent sequence x(n) in it; what must happen to the limit? (Remember that closed sets must contain the limits of their convergent sequences).
 

1. What does it mean for a subset of R^d to be closed?

A closed subset of R^d is one that contains all its boundary points. In other words, for every limit point of the subset, there exists a point within the subset that converges to it.

2. What is a countable intersection of closed subsets?

A countable intersection is a collection of subsets that are being intersected together, and the number of subsets in this collection is countably infinite. In this case, we are considering countable intersections of closed subsets of R^d.

3. Why is it important to prove that countable intersections of closed subsets of R^d are closed?

This proof is important because it is a fundamental property of topological spaces. It allows us to make generalizations and draw conclusions about countable intersections of closed subsets in any topological space, not just in R^d.

4. How can we prove that countable intersections of closed subsets of R^d are closed?

The proof involves showing that the limit point of the intersection of countably infinite closed subsets is contained within the intersection. This can be done using the definition of closed subsets and the properties of limit points.

5. Are there any exceptions to this property in other topological spaces?

No, this property holds true in any topological space. However, it is important to note that not all topological spaces have a countable basis, which is a necessary condition for this property to hold.

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