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 $\rho \subseteq X \times Y$, which means that it can be represented equivalently by its graph in the X-Y space. We write $x\rho\, y$ if $(x,y) \in \rho$ and define for $\rho$ its domain  ${\rm dom \ }\rho = \{ x \in X \mid \exists\, y\in Y \mbox{ with } (x,y)\in \rho \}$ and its image  ${\rm im \ }\rho = \{y \in Y \mid \exists\, x\in {\rm dom \ }\rho\mbox{ with } (x,y)\in \rho \}$. When ${\rm dom \ }\rho = X$ we say that $\rho$ is an entire  relation; we shall use only entire relations so we shall not need this qualification. A relation $\rho \subseteq X \times X$ may have any or none of the following properties:

uniqueness of image    
symmetry  		  $x\rho\, y$
if and only if $y \rho\, x$


reflexivity  		  for all $x \in X$,
$x\rho\, x$


transitivity  		  for all $x,y,z \in X$,
$x\rho\, y$
and $y \rho\,
z$
implies $x\rho\, z$


equivalence 		   symmetry, reflexivity, and transitivity 

antisymmetry  		  $x\rho\, y$
and $y \rho\, x$
implies x = y


partial order  		  antisymmetry, reflexivity, and transitivity.


total order  		  for all $x,y \in X$,
either $x\rho\, y$
or $y \rho\, x$

     
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 $x \in X$, there exists a unique $y\in Y$ that is related to the given x
Then we usually write y = f(x) or y = fx, and make the sets involved clear by

$$f:X \to Y:x\mapsto f(x).$$

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 $f:X \to Y$ 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)  		  ${\rm im \ }f = Y$;
denoted
                                                            $f:X\rightarrow\kern-.82em\rightarrow Y$


bijectivity (both)  		  injectivity and surjectivity 

The inclusion map of a subset $A \subseteq X$ is the map

$$i: A \hookrightarrow X:a\mapsto a.$$

The restriction of a map $f:X \to Y$ to a subset of its domain $A \subseteq X$ 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 $f:X \to Y$ is surjective, then we can always find a map $s: Y \to X$ such that $f \circ s = 1_Y$. 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

$$f:X \to Y:x\mapsto f(x)$$

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

$$f_{\rightarrow}:{\rm Sub}X \to {\rm Sub}Y:A\mapsto \{f(x)\vert \ x\in A\}$$

$$f^{\leftarrow}:{\rm Sub}Y \to {\rm Sub}X:B\mapsto \{x\in X\vert \ f(x)\in B\}$$

Normally, we just use f to denote what is really $f_{\rightarrow},$ but it is important to be very careful in writing f^-1 instead of $f^{\leftarrow},$ because whereas $f^{\leftarrow}$ always exists, f^-1 may not.