1)Prove that the intersection of an arbitrary collection of closed sets is closed. 2)Prove that the...

1)Prove that the intersection of an arbitrary collection of closed sets is closed.

2)Prove that the union of a finite collection of closed sets is closed

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Colby Messinger answered 1 year ago

Prove that the intersection of two compact sets is compact, using criterion (2).

Question: Prove that the intersection of two compact sets is compact, using criterion (2). Prove that the intersection of two compact sets is compact, using criterion (1). Prove that the intersection of two compact sets is compact, using criterion (3). Probably the most important new idea you'll encounter in real analysis is the concept of compactness. It's the compactness of [a, b] that makes a continuous function reach its maximum and that makes the Riemann in- tegral exist. For...

Provide an example 1) A nested sequence of closed, nonempty sets whose intersection is empty. 2)...

Provide an example 1) A nested sequence of closed, nonempty sets whose intersection is empty. 2) A set A that is not compact and an open set B such that A ∪ B is compact. 3) A set A that is not open, but removing one point from A produces an open set. 4) A set with infinitely many boundary points. 5) A closed set with exactly one boundary point

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Prove the following Theorems: 1. A finite union of compact sets is compact. 2. Any intersection...

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(a) Give an example of a collection of closed sets whose union is not closed. (b)...

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Let A, B, C be arbitrary sets. Prove or find a counterexample to each of the...

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Prove that for arbitrary sets A, B, C the following identities are true. Note that Euler...

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1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove...

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"•“"Suppose ℱ, ?1, and ?2 are nonempty families of sets. Prove that if ℱ ⊆ ?1...

"•“"Suppose ℱ, ?1, and ?2 are nonempty families of sets. Prove that if ℱ ⊆ ?1 ∩ ?2, then ∩?1 ∪ ∩?2 ⊆ ∩ℱ.“" Please explain what the question is asking for and break down the solution for me step by step with explanations please!

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