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Liad
Liad
17:06
@AlessandroCodenotti hi, free for a question? (in topo)
Alessandro Codenotti
Alessandro Codenotti
maybe
depends on how long the answer to it is
Liad
Liad
it is actually a general question about something im not sure :
Alessandro Codenotti
Alessandro Codenotti
ok
Liad
Liad
if $X$ , $Y$ topo. spaces than a basis for the product space is $\{ U x V : U $ is open in $X$ and $V$ is open in $Y \}$
if $B_X$ and $B_Y$ is bases for $X,Y$ than $\{B_1 x B_2 : B_1 \in B_X , B_2 \in B_Y \}$ is also a basis which is smaller
so far you are with me?
Alessandro Codenotti
Alessandro Codenotti
right, that works for finite products
(you can use \times in latex for a cartesian product symbol)
Liad
Liad
17:10
huh, ok. now the question is
the standard topo on $R \times R $ has basis , by def. , of the first form i said.
it is written in my book that the other basis (the second i wrote) is much smaller
it is only $(a,b) \times (c,d) $
why it is much smaller?
isn't it the same?
Alessandro Codenotti
Alessandro Codenotti
open sets in $\Bbb R$ are also things like $(a,b)\cup(c,d)$
Liad
Liad
because $U $ is open in $R$ means $U=(a,b)$
Alessandro Codenotti
Alessandro Codenotti
open sets in $\Bbb R$ are also things like $(a,b)\cup(c,d)$
so you get cartesian products of unions of interval rather than just product of intervals
Liad
Liad
but it is also open in $R \times R$
huh. got it.
ok
Alessandro Codenotti
Alessandro Codenotti
no, $(a,b)\times (c,d)$ is open in $\Bbb R^2$
Liad
Liad
17:15
thanks
Alessandro Codenotti
Alessandro Codenotti
you're welcome
Liad
Liad
im working on the second ex. now , i guess i will check with you my answers for the interesting questons tomorrow or latter on today :P
Alessandro Codenotti
Alessandro Codenotti
sure, I'll probably won't be there later today, but I'll read them when I can
Liad
Liad
17:34
@AlessandroCodenotti actually i have another question before i go to the ex.
Alessandro Codenotti
Alessandro Codenotti
17:58
What is it? @Liad
Liad
Liad
are you familiar with the dictionary order on $R \times R$ ?
Alessandro Codenotti
Alessandro Codenotti
More or less
Liad
Liad
a basis for this topo is $\{(a \times b , c \times d ): a<c \} \cup \{(a\times b , a\times c): b < c \}$ , am i correct?
Alessandro Codenotti
Alessandro Codenotti
Hm not sure
You want the order topology?
Liad
Liad
yes, with dictionary order
Alessandro Codenotti
Alessandro Codenotti
18:16
Hm, ok, so you need open intervals (x,y). Given x=(a,b) and y=(c,d) how are the elements in between them?
Liad
Liad
18:32
$\{(a\times b, c \times d ) \}$ for $a<c$ and $\{(a\times b , a\times d)\}$
this collections form a basis i think
 
3 hours later…
Liad
Liad
21:48
well it is. found it somewhere on the internet :P. almost finished the ex., 2 questions left. one seems interesting, will talk with you about it tomorrow.

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