A k-cell R is a subset of Rk of the form R= [a, b] x [a2,...

       

   

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A k-cell R is a subset of Rk of the form R= [a, b] x [a2, b₂] x [ak, bk], with <aj <b; <∞ for all 1 ≤j ≤k. Theorem 2.40 of the textbook shows that every k-cell is compact. Fill in the following sketch to arrive at an alternative proof of this fact. (a) Suppose first that k = 1. Let {Ga a € A} be an open cover of the 1-cell I = [a, b]. For a < x≤ b, set Iz = [a, z], and define the set X := {x € (a,b) : I, admits a finite subcover from {G}}. Show that X is nonempty and that sup(X) = b. Show that b is in fact the maximum element of X. Does this prove that I is compact? (b) Now suppose that k = 2, R = [a₁, b₁] × [a2, b₂], and {U} is an open cover of R. Define R= [a₁,2] x [a2, b2]. Use part (a) of this problem to show that the set x := {2 € (a1, b₁] : R₂ admits a finite subcover from {U.}} has b₁ as its maximum, proving that R is compact. (c) Generalize the argument above to show that a k-cell is compact for any k € N. 2.4 Definition For any positive integer n, let J, be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers. For any set A, we say: (a) A is finite if A~J, for some n (the empty set is also considered to be finite). (b) (c) A is countable if A~ J. (d) A is uncountable if A is neither finite nor countable. (e) A is at most countable if A is finite or countable. A is infinite if A is not finite. Countable sets are sometimes called enumerable, or denumerable. For two finite sets A and B, we evidently have A B if and only if A and B contain the same number of elements. For infinite sets, however, the idea of "having the same number of elements" becomes quite vague, whereas the notion of 1-1 correspondence retains its clarity. A k-cell R is a subset of Rk of the form R= [a, b] x [a2, b₂] x [ak, bk], with <aj <b; <∞ for all 1 ≤j ≤k. Theorem 2.40 of the textbook shows that every k-cell is compact. Fill in the following sketch to arrive at an alternative proof of this fact. (a) Suppose first that k = 1. Let {Ga a € A} be an open cover of the 1-cell I = [a, b]. For a < x≤ b, set Iz = [a, z], and define the set X := {x € (a,b) : I, admits a finite subcover from {G}}. Show that X is nonempty and that sup(X) = b. Show that b is in fact the maximum element of X. Does this prove that I is compact? (b) Now suppose that k = 2, R = [a₁, b₁] × [a2, b₂], and {U} is an open cover of R. Define R= [a₁,2] x [a2, b2]. Use part (a) of this problem to show that the set x := {2 € (a1, b₁] : R₂ admits a finite subcover from {U.}} has b₁ as its maximum, proving that R is compact. (c) Generalize the argument above to show that a k-cell is compact for any k € N. 2.4 Definition For any positive integer n, let J, be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers. For any set A, we say: (a) A is finite if A~J, for some n (the empty set is also considered to be finite). (b) (c) A is countable if A~ J. (d) A is uncountable if A is neither finite nor countable. (e) A is at most countable if A is finite or countable. A is infinite if A is not finite. Countable sets are sometimes called enumerable, or denumerable. For two finite sets A and B, we evidently have A B if and only if A and B contain the same number of elements. For infinite sets, however, the idea of "having the same number of elements" becomes quite vague, whereas the notion of 1-1 correspondence retains its clarity.