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srfriggen
#14
Sep23-11, 06:13 PM
P: 258
Quote Quote by micromass View Post
The key to the upper bound is the function

[tex]\mathcal{C}(\mathbb{R},\mathbb{R})\rightarrow \mathbb{R}^\mathbb{Q}:f\rightarrow f\vert_\mathbb{Q}[/tex]

it suffices to show that this function is an injection (since [itex]\mathbb{R}^\mathbb{Q}[/itex] has cardinality c). For this, we must show that if [itex]f\vert_\mathbb{Q}=g\vert_\mathbb{Q}[/itex], then f=g. This holds since [itex]\mathbb{Q}[/itex] is countable.
Perhaps you can just help me with some of the terminology... I'm sorry but I simply don't understand what a lot of the symbols you used are. I've never encountered functions written with a set as a superscript or a subscript. Does the "C" mean cardinality? Also, I'm assuming "dense" is equivalent to "countable"