Question #140050
Find a countable dense subset of (C[0,1],ρ) where ρ denotes the uni-form metric.(You do not need to justify)
1
Expert's answer
2020-10-25T19:09:48-0400

Let us denote P\mathscr{P} is the restriction of all polynomial in [0,1][0,1] .

Now, define

A:={PP:Coefficients of P are rational number}\mathscr{A}:=\{P\in\mathscr{P}:\text{Coefficients of P are rational number}\}

A\mathscr{A} is dense in C[0,1]\mathscr{C}[0,1] .


Though you did not asked for proof still I am giving Hints for the proof:

Note that A\mathscr{A} is countable and Then use Weierstrass approximation theorem.


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