Prove:If $(X,\tau)$ is a separable Hausdorff space ,then there are at most $c$ distinct continuous functions $f:(X,\tau)\to[0,1]$.

Question

If $(X,\tau)$ is a separable Hausdorff space ,then there are at most $c$ distinct continuous functions $f:(X,\tau)\to[0,1]$.

Stuck for days. Please help.

Answer 1

Let $D$ be a countable dense subset of $X$. Show that the map $f \in C(X, [0,1]) \to f|D \in [0,1]^D$ is injective (1-1). So then

$$|C(X, [0,1])| \le |[0,1]^{\aleph_0} = \mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = \mathfrak{c}$$

Answer 2

Separable means having a countable dense subset. If $A$ is a countable dense subset of $X$ then two continuous maps $f$, $g:X\to[0,1]$ agreeing on $A$ must agree on all of $X$ (why?). So the number of continuous maps $X\to[0,1]$ is at most the number of all maps $A\to[0,1]$ etc.