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Principles of Mathematical Analysis, Third Edition

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Basic Topology Finite, Countable, and Uncountable Sets 2.1 Definition If A and B are sets, a function f from A into B is a subset of A × B such that for every a in A there exists a unique b in B such that ( a, b ) f . The uniqueness condition can be formulated explicitly as follows: if ( a, b ) f and ( a, c ) f , then b = c . We call A the domain , and B the codomain , of f . The function f from A to B is usually denoted f : A B , and for each a A , the unique b B such that ( a, b ) f is denoted by f ( a ). 2.2 Definition Let A and B be two sets and let f be a mapping of A into B . If E A , the image of E under f is defined as f ( E ) = { f ( x ): x E } . The range of f is f ( A ). If f ( A ) = B , we say that f maps onto B . If C B , the pre-image of C under f is defined as f 1 ( C ) = { x : f ( x ) C } . We say that f is one-to-one if x 1 negationslash = x 2 implies that f ( x 1 ) negationslash = f ( x 2 ) for all x 1 , x 2 A . 2.3 Definition If there exists a one-to-one mapping of A onto B , we say that A and B can be put in one-to-one correspondence , or that A and B have the same cardinal number , or briefly, that A and B are equivalent , and we write A B . One-to-one correspondence is an equivalence relation: It is reflexive: A A . It is symmetric: If A B , then B A . It is transitive: If A B and B C , then A C . 2.4 Definition For any positive integer n , let J n 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 n for some n ; the empty set is considered finite. (b) A is infinite if A is not finite. (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. 2.7 Definition By a sequence , we mean a function f defined on the set J of all positive integers If f ( n ) = x n , for n J , it is customary to denote the sequence f by the symbol { x n } , or sometimes by x 1 , x 2 , x 3 , . . . . 2.8 Theorem Every infinite subset of a countable set A is countable. 2.9 Definition Let A and Ω be sets, and let { E α } be a family of subsets of Ω with index A . The union of the sets E α is defined to be the set S such that x S if and only if there exists an α A such that x E α . We write S = uniondisplay α A E α , or S = n uniondisplay k =1 E n if A = J n , or S = uniondisplay k =1 E n if A = J.
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ch2 - Basic Topology Finite Countable and Uncountable Sets...

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