ch2

Principles of Mathematical Analysis, Third Edition

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 ....
View Full Document

This document was uploaded on 05/22/2008.

Page1 / 5

ch2 - Basic Topology Finite, Countable, and Uncountable...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online