Give an example of an open cover of (0, 1) that contains no finite subcover of (0, 1). Give an example of an open cover of (0,1) that contains no finite subcover of (0,1).
Give an example of an open cover of (0,1) that contains no finite subcover of (0,1).
4. Let Uαα∈A be a finite open cover of a compact metric space X. For question for (a), (b) Remark: ε is called a Lebesgue number of the cover. (a) Show that there exists ε>0 such that for each x∈X, the open ball B(x;ε) is contained in one of the Uα’s. (b) Show that if at least one of the Uα’s is a proper subset of X, then there is a largest Lebesgue number for the cover. 4. Let {U}aea...
2. Consider the set S-[1, oo). Consider the open cover x(n-1,n+)InEN) - (0,2),(1,3),(2.4),(8,5.,..) of S. Prove that X contains no finite subcover of S. Hence S is not compact. 2. Consider the set S-[1, oo). Consider the open cover x(n-1,n+)InEN) - (0,2),(1,3),(2.4),(8,5.,..) of S. Prove that X contains no finite subcover of S. Hence S is not compact.
Carefully and rigorously prove the following. Let X be a metric space. Show X is compact if and only if every sequence contains a convergent subse- quence. Hint for (): Argue by contradiction. If there was a sequence with no convergent subsequence, use that sequence to construct an open cover of X, such that every set in the cover contains only a finite number of elements of the sequence. Then use compactness to get a contradiction. Hint for (): Let...
III. (15 pts total) a) Define what it means for a set SCR to be compact. b) Consider the set S = {CERO <<<1}. Give an example of an open cover of S that has a finite subcover. c) With S as above, show that S is not compact by giving an ex- ample of an open cover that does not admit a finite subcover. Justify your answer.
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number α ∈ [0, 1] (as a subset of the reals now!) and for each (rational) q ∈ [0, 1] look for an...
Find the maximum value of Max F(x, y) = 4 x y ; X>0 y y so Subject to : x² + y = 1 an 16
4. Find a conformal equivalence from the open unit disc to the set W = {z : 0 < arg(z) < π/4)