Help understanding Proof of Konig's Theorem

I've come across Konig's Theorem in Jech, and I'm having some trouble understanding the proof which can be found here. I don't understand what Jech means by S_i, the projection into the ith coordinate. Can someone give a basic example so I can wrap my head around it? Thanks.

For a very simple example, suppose that $I=\{0,1,2\}$. Let $T_0,T_1$, and $T_2$ be any sets, let $T=\prod_{i\in I}T_i$, and let $Z\subseteq T$. By the definition of product, the elements of $T$ are the functions $f$ with domain $I$ such that $f(i)\in T_i$ for each $i\in I$. For example, if $T_0=\{0,1\}$, $T_1=\{2,3\}$, and $T_2=\{4,5\}$, there are exactly $2^3=8$ functions in $T$:

\begin{align*} &0\mapsto 0,1\mapsto 2,2\mapsto 4\\ &0\mapsto 0,1\mapsto 2,2\mapsto 5\\ &0\mapsto 0,1\mapsto 3,2\mapsto 4\\ &0\mapsto 0,1\mapsto 3,2\mapsto 5\\ &0\mapsto 1,1\mapsto 2,2\mapsto 4\\ &0\mapsto 1,1\mapsto 2,2\mapsto 5\\ &0\mapsto 1,1\mapsto 3,2\mapsto 4\\ &0\mapsto 1,1\mapsto 3,2\mapsto 5 \end{align*}

(Of course in the proof we’re dealing with much larger sets.)

If $f$ is any member of $T$, and $i\in I$, then by definition the projection of the single function $f$ on the $i$-th coordinate is $f(i)$. In our toy example suppose that $f$ is the function

$$0\mapsto 0,1\mapsto 3,2\mapsto 5\;;$$

the projection of $f$ to coordinate $0$ is $f(0)=0$, the projection of $f$ to coordinate $1$ is $f(1)=3$, and the projection of $f$ to coordinate $2$ is $f(2)=5$.

Now suppose that we have a set of such functions, i.e., a set $Z\subseteq T$; we might use the example $Z=\{f,g,h\}$, where $f$ is

$$0\mapsto 0,1\mapsto 3,2\mapsto 5\;,$$

$g$ is

$$0\mapsto 1,1\mapsto 2,2\mapsto 5\;,$$

and $h$ is

$$0\mapsto 1,1\mapsto 3,2\mapsto 4\;.$$

The projection of $Z$ to the $0$ coordinate is just $\{f(0),g(0),h(0)\}=\{0,1\}$, the set of projections to the $0$ coordinate of the members of $Z$.

Similarly, the projection of $Z$ to the $1$ coordinate is $\{f(1),g(1),h(1)\}=\{2,3\}$, and the projection of $Z$ to the $2$ coordinate is $\{f(2),g(2),h(2)\}=\{4,5\}$.

Projection here can be thought of in geometric terms. If $I=\{x,y\}$, and $T_x=T_y=\Bbb R$, then $T$ is essentially just $\Bbb R^2$, the plane. If $Z\subseteq\Bbb R^2$, we can project $Z$ onto the $x$-axis by forming the set of all $x$-coordinates of points of $Z$, as if we were using the $x$-axis as a projection screen. Similarly, the set of $y$-coordinates of points of $Z$ is the projection of $Z$ to the $y$-axis.

In my example I took the projections of a single subset of $T$ to each of the coordinates. In the proof of König’s theorem you actually have a set $Z_i$ for each coordinate $i\in I$, and you project the set $Z_i$ only to the $i$-th coordinate. In geometric terminology you have one set, $Z_i$ for each coordinate axis (each $i\in I$), and you project each of those sets to a different axis.

• Wow thank you. This is extremely clear and exactly what I was looking for. – user2034 Feb 17 '15 at 14:06
• @user2034: You're welcome; glad it helped. – Brian M. Scott Feb 17 '15 at 18:04