Answer to Question #140050 in Differential Geometry | Topology for Askin

Question

Find a countable dense subset of (C[0,1],ρ) where ρ denotes the uni-form metric.(You do not need to justify)

Answer 1

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

Now, define

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

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

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

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