Construct an explicit bijection g: [0,1] --> (0,1).
Hint: One way to do it would be three parts. One for g(0), one for g(1/n) for n belongs to N and one for everything else.
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Construct an explicit bijection g: [0,1] --> (0,1).
Hint: One way to do it would be three parts. One for g(0), one for g(1/n) for n belongs to N and one for everything else.
Does the following picture give you any ideas?
https://lh4.googleusercontent.com/-K.../bijection.png
A little bit. I'm still not sure how to write it.
DefineQuote:
Originally Posted by allstar2
,
define
and elsewise
.
Thank you!
A pretty standard example is this: the set of all rational numbers in (0, 1) is countable so can be "listed". Map 0 to
, 1 to
, and for all rational numbers
to
. All irrational numbers in (0, 1) are mapped to themselves.