• Jan 11th 2009, 08:48 PM
Mathstud28
I started this thread to post questions about words...not math :P.

I understand what this means $\displaystyle \pi_j:A\times B\longmapsto B$ but they call it a projection map. Is there something else that goes along with $\displaystyle \pi_j:A\times B\longmapsto B$ than meets the eyes?
• Jan 11th 2009, 11:26 PM
Moo
I'm not sure I understand what you're asking for... Are you wondering if there is an application of this ? (Surprised)

I can show you one (the first that came in my mind), but it has to do with measure theory (product sigma-algebra). So if you're interested, I can show it...
And Opalg once used them to explain stuff in product topology.

There another way of seeing these functions, it's calling them the coordinates mappings :
$\displaystyle \pi_1 ~:~ A \times B \to A$ (note that it's $\displaystyle \to$ and not $\displaystyle \mapsto$, I don't know how you learnt it, but my teachers always used the second one for defining the function)

$\displaystyle (x_1,x_2) \mapsto x_1$

$\displaystyle \pi_2 ~:~ A \times B \to B$
$\displaystyle (x_1,x_2) \mapsto x_2$
• Jan 12th 2009, 03:22 PM
Mathstud28
Quote:

Originally Posted by Moo
I'm not sure I understand what you're asking for... Are you wondering if there is an application of this ? (Surprised)

I can show you one (the first that came in my mind), but it has to do with measure theory (product sigma-algebra). So if you're interested, I can show it...
And Opalg once used them to explain stuff in product topology.

There another way of seeing these functions, it's calling them the coordinates mappings :
$\displaystyle \pi_1 ~:~ A \times B \to A$ (note that it's $\displaystyle \to$ and not $\displaystyle \mapsto$, I don't know how you learnt it, but my teachers always used the second one for defining the function)

$\displaystyle (x_1,x_2) \mapsto x_1$

$\displaystyle \pi_2 ~:~ A \times B \to B$
$\displaystyle (x_1,x_2) \mapsto x_2$

Thank you very much Moo, I found what they meant and it is as I suspected and you of course identified...it was meant to be a coordinate mapping...btw they do use $\displaystyle \pi_1:A\times B{\color{red}\longmapsto}B$