Proof That the Continuous Functions on [0,1] With the Maximum Metric are Second Countable

Theorem

The setof continuous functions onwith the maximum metricis second countable.

Proof

LetAccording to the Weierstrass approximation theorem, there existas a polynomial with rational coefficients such thatfor all

Hence the set of polynomials with rational coefficients is dense in C[0,1].

The set of polynomials with rational coefficients is countable hence C[0,1] contains a countable dense subset P[0,1] which is separable. A sepable metric space is second countable so C[0,1] is second countable with the maximum metric.