MATH 5345H — Week 2: Cartesian products; finite, countable, and uncountable sets

0.1 (§5) Cartesian products

•
Def. Indexing function for $\mathcal{A}$ : a surjection $f\colon J\to\mathcal{A}$ ; write $A_{\alpha}=f(\alpha)$ , family $\{A_{\alpha}\}_{\alpha\in J}$ . Not assumed injective: $A_{\alpha}=A_{\beta}$ is allowed for $\alpha\neq\beta$ .
•
Notation. $\bigcap_{\alpha\in J}A_{\alpha}$ , $\bigcup_{\alpha\in J}A_{\alpha}$ ; finite case $A_{1}\cap\cdots\cap A_{n}$ .
•
Def. $n$ -tuple $=$ function $x\colon\{1,\dots,n\}\to X$ , written $(x_{1},\dots,x_{n})$ ; $\prod_{i=1}^{n}A_{i}$ ; $X^{n}$ .
•
Def. sequence $=$ function $x\colon\mathbb{N}\to X$ , written $(x_{i})_{i=1}^{\infty}$ (also called an $\omega$ -tuple); $\prod_{i=1}^{\infty}A_{i}=A_{1}\times A_{2}\times\cdots$ ; $X^{\omega}$ .
•
$\bigstar$ Def. General product. $J$ -tuple $=$ function $x\colon J\to X$ , $x_{\alpha}=x(\alpha)$ .
$\prod_{\alpha\in J}A_{\alpha}=\Bigl{\{}\,x\colon J\to\bigcup_{\alpha\in J}A_{% \alpha}\ \Bigm{|}\ x(\alpha)\in A_{\alpha}\ \forall\alpha\,\Bigr{\}},\qquad X^% {J}=\prod_{\alpha\in J}X.$
Note. Stress: a point of an infinite product is a function. Everything about product topologies later reads more easily from this description.

•
Def. Section of $\mathbb{N}$ : $\{1,2,\dots,n\}$ ; for $n=0$ this is $\emptyset$ .
•
Lem. $\{1,\dots,m\}\hookrightarrow\{1,\dots,n\}$ injective $\Rightarrow m\leq n$ .
Proof. Induction on $n$ : delete $f(m)=k$ , use a bijection $\{1,\dots,n\}-\{k\}\to\{1,\dots,n-1\}$ .
•
Cor. Hence there is no injective $\{1,\dots,m\}\to\{1,\dots,n\}$ when $m>n$ . (Pigeonhole.)
•
Prop. A bijection $\{1,\dots,m\}\to\{1,\dots,n\}$ forces $m=n$ .
Proof. Apply the lemma to $f$ and to $f^{-1}$ .
•
Cor. Hence there is no bijection $\{1,\dots,m\}\to\{1,\dots,n\}$ when $m\neq n$ .
•
Def. $A$ is finite of cardinality $n$ if there is a bijection $A\to\{1,\dots,n\}$ . Cardinality is well defined. Ex. $\emptyset$ has cardinality $0$ ; singletons have cardinality $1$ .
•
Lem. $A\subset\{1,\dots,n\}$ is finite, of cardinality $\leq n$ ; if $A\subsetneq\{1,\dots,n\}$ the cardinality is $<n$ .
•
$\bigstar$ Thm. A finite set admits no bijection with a proper subset of itself.
•
Cor. $\mathbb{N}$ is not finite: $f(x)=x+1$ is a bijection of $\mathbb{N}$ with the proper subset $\mathbb{N}-\{1\}$ .
•
Cor. Any subset $B$ of a finite set $A$ is finite; if $B\subsetneq A$ then $\mathrm{card}(B)<\mathrm{card}(A)$ .
•
Prop. TFAE: (1) $A$ finite; (2) some $\{1,\dots,n\}\twoheadrightarrow A$ ; (3) some $A\hookrightarrow\{1,\dots,n\}$ .
Proof. $(2)\Rightarrow(3)$ : send $x$ to $\min g^{-1}(x)$ .
•
Prop. Finite unions and finite products of finite sets are finite.