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