- #1
Bachelier
- 376
- 0
I don't understand this point:
Given the open set E = U_(a in L) I_a. Union of open intervals
We're showing this is countable.
WTS is that indexed set L is countable.
Set g: L---> Q (rationals) because Q is dense then every interval meets Q.
a---> q_a
this is 1-1. But here's where I get confused:
The conclusion is: Hence L is bijectively equivalent to a subset of Q and hence is countab.
I understand the fact that a subset of countable set is countable. But g codomain is Q. Should I restrict it to some Q_a in Q and show that g is invertible. Hence bij.
We only proved g is 1-1. Where did the bij. argument come from.
Thanks guys
Given the open set E = U_(a in L) I_a. Union of open intervals
We're showing this is countable.
WTS is that indexed set L is countable.
Set g: L---> Q (rationals) because Q is dense then every interval meets Q.
a---> q_a
this is 1-1. But here's where I get confused:
The conclusion is: Hence L is bijectively equivalent to a subset of Q and hence is countab.
I understand the fact that a subset of countable set is countable. But g codomain is Q. Should I restrict it to some Q_a in Q and show that g is invertible. Hence bij.
We only proved g is 1-1. Where did the bij. argument come from.
Thanks guys