chapter3 - Chapter 3 METRIC SPACES and SOME BASIC TOPOLOGY...

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Chapter 3 METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under- standing and ability to make use of properties of U ± U 1 . The next goal is to generalize our work to U n and, eventually, to study functions on U n . 3.1 Euclidean n -space The set U n is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. For completeness, we include the following De ¿ nition 3.1.1 Let S and T be sets. The Cartesian product of S and T , denoted by S ² T, is ³ ± p ² q ³ : p + S F q + T ´ . The Cartesian product of any ¿ nite number of sets S 1 ² S 2 ² ´´´² S N , denoted by S 1 ² S 2 ² µ µ µ ² S N , is j ± p 1 ² p 2 ² ´´´² p N ³ : ± 1 j ³ b ± j + M F 1 n j n N ³ " p j + S j ck . The object ± p 1 ² p 2 ² ´´´² p N ³ is called an N -tuple . Our primary interest is going to be the case where each set is the set of real numbers. 73
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74 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY De ¿ nition 3.1.2 Real n -space , denoted U n , is the set all ordered n-tuples of real numbers ± i.e., U n ± ³ ± x 1 ² x 2 ² ´´´² x n ³ : x 1 ² x 2 ² ´´´² x n + U ´ . Thus, U n ± U ² U ² µ µ µ ² U _ ^] n of them , the Cartesian product of U with itself n times. Remark 3.1.3 From MAT108, recall the de ¿ nition of an ordered pair : ± a ² b ³ ± def ³³ a ´ ² ³ a ² b ´´ . This de ¿ nition leads to the more familiar statement that ± a ² b ³ ± ± c ² d ³ if and only if a ± b and c ± d. It also follows from the de ¿ nition that, for sets A, B and C, ± A ² B ³ ² C is, in general, not equal to A ² ± B ² C ³ ± i.e., the Cartesian product is not associative. Hence, some conventions are introduced in order to give meaning to the extension of the binary operation to more that two sets. If we de ¿ ne ordered triples in terms of ordered pairs by setting ± a ² b ² c ³ ± ±± a ² b ³² c ³ ± this would allow us to claim that ± a ² b ² c ³ ± ± x ² y ² z ³ if and only if a ± x, b ± y, and c ± z. With this in mind, we interpret the Cartesian product of sets that are themselves Cartesian products as “big” Cartesian products with each entry in the tuple inheriting restrictions from the original sets. The point is to have helpful descriptions of objects that are described in terms of n-tuple. Addition and scalar multiplication on n -tuple is de ¿ ned by ± x 1 ² x 2 ² ´´´² x n ³ ± y 1 ² y 2 ² ´´´² y n ³ ± ± x 1 y 1 ² x 2 y 2 ² ´´´² x n y n ³ and : ± x 1 ² x 2 ² ´´´² x n ³ ± ±: x 1 ² : x 2 ² ´´´² : x n ³ , for : + U , respectively. The geometric meaning of addition and scalar multiplication over U 2 and U 3 as well as other properties of these vector spaces was the subject of extensive study in vector calculus courses (MAT21D on this campus). For each n , n o 2, it can be shown that U n is a real vector space.
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