Prove that the union of a finite collection of compact subsets is compact

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Colby Messinger answered 2 years ago

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

Prove the following Theorems: 1. A finite union of compact sets is compact. 2. Any intersection of compact set is compact. 3. A closed subset of a compact set is compact. 4. Every finite set in IRn is compact.

Prove that the product of a finite number of compact spaces is compact.

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N...

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N such that for all n, Kn+1 ⊂ Kn. Show that K = T∞ n=1 Kn is compact. Can K ever be the empty set?

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection...

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection of vectors in a vector space V and suppose that it is not a linearly independent set. i. Show that one can find a vector w ∈ {v1, . . . , vn} such that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that Span(S) = Span(v1, . . . , vn). ii. Suppose T ⊂ {v1,...

Prove the following stronger variant of Proposition 7.4. Suppose C is collection of connected subsets of...

Prove the following stronger variant of Proposition 7.4. Suppose C is collection of connected subsets of a metric space X and B ∈ C. Show, if for each A ∈ C, A ∩ B not equal ∅, then Γ = ∪{C : C ∈ C} is connected. [Suggestion: Consider the collection D = {C ∪ B : C ∈ C}].

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?....

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?. Prove ?(⋂ ?? ?∈? ) ⊂ ⋂ ?(??) ?∈? with equality if ? is one-to-one. 2. Compute: a. ⋂ ∞ ?=1 [?,∞) b. ⋃ ∞ ?=1 [0,2 − 1 /?] c. lim sup ?→∞ (−1 + (−1)^? /?,1 +(−1)^? /?) d. lim inf ?→∞(−1 +(−1)^?/ ?,1 +(−1)^? /?)

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...

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Prove that the union of a finite collection of compact subsets is compact

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