ONT Re: Topology
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Note 27
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| 3. Product and Quotient Spaces
|
| 3.1. Continuous Functions
|
| For convenience we review some of the terminology and a
| few elementary propositions about functions (chapter 0).
| The words "function", "map", "mapping", "correspondence",
| "operator", and "transformation" are synonymous.
|
| A function f is said to be on X iff its domain is X. It is to Y,
| or into Y, iff its range is a subset of Y, and it is onto Y iff
| its range is Y. The value of f at a point x is f(x), and f(x)
| is also called the image under f of x.
|
| If B is a subset of Y, then the inverse under f of B, f^(-1)[B],
| is {x : f(x) in B}. The inverse under f of the intersection (union)
| of the members of a family of subsets of Y is the intersection (union)
| of the inverses of the members; that is, if Z_c is a subset of Y for
| each member c of a set C, then:
|
| f^(-1)[ |^| {Z_c : c in C} ] = |^| {f^(-1)[Z_c] : c in C},
|
| and similarly for unions.
|
| If y is in Y, then f^(-1)[{y}], the inverse of the
| set whose only member is y, is abbreviated f^(-1)[y].
|
| The image f[A] of a subset A of X is the set of
| all points y such that y = f(x) for some x in A.
|
| The image of the union of a family of subsets of X is
| the union of the images, but, in general, the image of
| the intersection is not the intersection of the images.
|
| A function is one to one iff no two distinct points have
| the same image, and in this case f^(-1) is the function
| inverse to f.
|
| ( Notice that the notation is arranged so that,
| roughly speaking, square brackets occur in the
| designations of subsets of the range and domain
| of a function, and parentheses in the designations
| of members. For example, if f is one to one onto Y
| and y in Y, then f^(-1)(y) is the unique point x of X
| such that f(x) = y, and f^(-1)[y] = {x}.)
|
| JLK, Gen Top, pages 84-85.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤