*This preview shows
pages
1–2. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**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