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# Sets and maps

We recall some basic definitions. Definitions
Let X and Y be non-empty sets. A relation  from X to Y is a subset , which means that it can be represented equivalently by its graph in the X-Y space. We write if and define for its domain  and its image  . When we say that is an entire  relation; we shall use only entire relations so we shall not need this qualification. A relation may have any or none of the following properties:

uniqueness of image
symmetry
if and only if

reflexivity  		  for all ,

transitivity  		  for all ,

and
implies

equivalence 		   symmetry, reflexivity, and transitivity
antisymmetry
and
implies x = y

partial order  		  antisymmetry, reflexivity, and transitivity.

total order  		  for all ,
either
or



A function or map from a set X to a set Y is a set of ordered pairs from X and Y (pairs like (x,y) are the coordinates in the graph of the function) satisfying the uniqueness of image property:
for all , there exists a unique that is related to the given x
Then we usually write y = f(x) or y = fx, and make the sets involved clear by

Note that a map is equivalent to its graph, as a set of ordered pairs of coordinates in X-Y space; the graph of a map must not pass twice through any point in its domain--unlike a general relation. A map may have any or none of the following properties:

uniqueness of image
injectivity (1 to 1)  		  f(x) = f(y) implies x=y
surjectivity (onto)  		  ;
denoted

bijectivity (both)  		  injectivity and surjectivity


The inclusion map of a subset is the map

The restriction of a map to a subset of its domain is the composite map f|A = fi. The Axiom of Choice  is required for a number of constructions in topology and a convenient form is this:
Every surjection has a right inverse.
That is, if is surjective, then we can always find a map such that . Then s is called a section  of f. Equivalently, given any collection (not necessarily countable) of sets, it is possible to choose one element from each. Given a map

we get free two maps on subsets, one going each way:

Normally, we just use f to denote what is really but it is important to be very careful in writing f-1 instead of because whereas always exists, f-1 may not.

Next: Topology Up: Introducing Knots Previous: Introducing Knots
Kit Dodson
2000-01-24