# Projection map

• Feb 11th 2009, 02:51 PM
HTale
Projection map
I am stuck by this question,

"In general show that for all $\displaystyle j \in I$ there is a map

$\displaystyle \pi_j : \prod_{i \in I} X_i \rightarrow X_j$

which is the 'projection onto the j-th factor' ".

I'm stuck because I didn't even know you could ask this question; I'm all at sea about how to even do this!

Note, that $\displaystyle \pi_j : A_1 \times A_2 \times \cdots \times A_j \times \cdots \times A_n \rightarrow A_j$, given by $\displaystyle \pi_j(a_1, a_2, \ldots, a_j, \ldots, a_n) = a_j$
• Feb 12th 2009, 09:28 AM
Moo
Hello,
Quote:

Originally Posted by HTale
I am stuck by this question,

"In general show that for all $\displaystyle j \in I$ there is a map

$\displaystyle \pi_j : \prod_{i \in I} X_i \rightarrow X_j$

which is the 'projection onto the j-th factor' ".

I'm stuck because I didn't even know you could ask this question; I'm all at sea about how to even do this!

Note, that $\displaystyle \pi_j : A_1 \times A_2 \times \cdots \times A_j \times \cdots \times A_n \rightarrow A_j$, given by $\displaystyle \pi_j(a_1, a_2, \ldots, a_j, \ldots, a_n) = a_j$

I'm not sure I grabbed the problem correctly...

Let's assume $\displaystyle I=\{i_1,i_2,\dots,i_j,\dots \}$ (a countable set, otherwise it makes no sense to write the product)
Let $\displaystyle x \in \prod_{i \in I} X_i$.
Then x can be written as a tuple :
$\displaystyle x=(x_{i_1},x_{i_2},x_{i_3},\dots,x_{i_j},\dots)$, where $\displaystyle x_{i_k} \in X_{i_k}$ for any $\displaystyle i_k \in I$

$\displaystyle n \in I$ means that there exists $\displaystyle j$ such that $\displaystyle n=i_j$

And $\displaystyle \pi_n$ will yield the $\displaystyle i_j$-th coordinate of x, that is $\displaystyle x_{i_j}$

So it's defined... ?