Function (notation)

This page describes the notation of how we use functions for information on what a function is, see function

Basics

• $f:X\rightarrow Y$ - the most basic form, here $f$ is a relation that associates with each $x\in X$ a $y\in y$. We write this as $y=f(x)$

Abuses of notation

• Tuples: sometimes we will write $f:(X,A)\rightarrow(Y,B)$, this simply means that $f:X\rightarrow Y$ where $X$ is some sort of space (with structure $A$) and $Y$ is some sort of space with a structure $B$.
  • Possible misinterpretation:

    $f:X\times A\rightarrow Y\times B$ denotes a function, $f$ that takes ordered pairs, $(x,a)\in X\times A$ to ordered pairs, $(y,b)\in Y\times B$, this notation clearly operates on sets (as it uses the Cartesian product) keeping with the convention of the thing either side of the $\rightarrow$ is a set. So the notation $f:(X,A)\rightarrow(Y,B)$ for $f$'s input being a tuple, $(x,a)$ is absurd because:

    1. It is another notation for something we already have
    2. It violates the "sets being either side of the arrow" thing ($A\times B$ is a set, $(\cdot,\cdot)$, even if considered as an Ordered pair does not "evaluate" to something useful when it comes to relations.

  • Warnings:
    1. Sometimes $f:(X,A)\rightarrow(Y,B)$ denotes that $f:X\rightarrow Y$ with the additional information of $f\vert_A:A\rightarrow B$, or more simply is to say that $f:X\rightarrow Y$ with the additional statement: $f(A)\subseteq B$
       • Use this if $(X,A)$ has not previously been declared as some sort of space.
       • I've only ever seen this used in one book - Fundamentals of Algebraic Topology by Steven H. Weintraub
  • Examples:
    1. Let $(X,\mathcal{J})$ and $(Y,\mathcal{K})$ be topological spaces, let $f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$ be a continuous map...
       • Here the tuples (as usual) help the reader/writer keep track of spaces, in this case the topologies on $X$ and $Y$
       • This example extends to measurable spaces, vector spaces and many more.