By Patrick M. Fitzpatrick

Simply grasp the elemental strategies of mathematical research with complicated CALCULUS. offered in a transparent and straightforward method, this complex caluclus textual content leads you to an exact realizing of the topic by means of supplying you with the instruments you must prevail. a large choice of routines is helping you achieve a real figuring out of the cloth and examples show the importance of what you research. Emphasizing the team spirit of the topic, the textual content indicates that mathematical research isn't really a set of remoted proof and methods, yet quite a coherent physique of data.

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**Example text**

In the partially ordered set N × N with the ordering ( j, k) ≤ (m, n) iff j ≤ m and k ≤ n, consider the sequence (n, n), for n = 0, 1, 2, . . Describe all the maximal chains that include the given sequence. Do any of these chains have upper bounds? 4. Find a partially ordered set (X, ≤), with as few elements as possible, for which the hypothesis of Zorn’s lemma holds (every chain has an upper bound) but which does not have a maximum. 5. Show, assuming AC, that any Cartesian product of finite sets is either finite or uncountable (can’t be countably infinite).

Proof. (1) implies (2): suppose f (x) ∈ U ∈ U . Then x ∈ f −1 (U ), so for some j, xi ∈ f −1 (U ) for all i > j. Then f (xi ) ∈ U , so f (xi ) → f (x). (2) implies (3): let F → x. If f [[F ]] → f (x) (that is, f [[F ]] does not converge to f (x)), take f (x) ∈ U ∈ U with f [A] ⊂ U for all A ∈ F . Define a partial ordering on F by A ≤ B iff A ⊃ B for A and B in F . By definition of filter base, (F , ≤) is then a directed set. Define a net (using AC) by choosing, for each A ∈ F , an x(A) ∈ A with f (x(A)) ∈ / U .

If h is 1–1, it is a homeomorphism. Proof. 3. Let h −1 (A) be open. 2. 9, so A is open. If f ◦ h is continuous, then for any open U ⊂ S, ( f ◦ h)−1 (U ) = h −1 ( f −1 (U )) is open. So f −1 (U ) is open and f is continuous. 10. The power set 2 X , which is the collection of all subsets of a set X , can be viewed via indicator functions as the set of all functions from X into Problems 41 {0, 1}. In other words, 2 X is a Cartesian product, indexed by X , of copies of {0, 1}. With the usual discrete topology on {0, 1}, the product topology on 2 X is compact.