Properties of Cp(X) In class, we learned about the space $C_p(X)$, which is the space of all continuous real-valued functions on $X$ with the topology of point-wise convergence. To better understand the material, I started looking at problems. The one problem I found that I am having trouble with was asked to verify properties of $C_p(X)$. It goes as follows:
Consider the space $X = C_p(I)$, where $I = [0,1]$ Show the following:
(1) $X$ is not metrizable
(2) $X$ does not have a countable base
(3) $X$ is not first-countable
(4) $X$ is separable
(5) every one-point subset of $X$ is a $G_{\delta}$-set
I keep finding sources that use cardinal functions, but we never talked about them in class. How would this be done not using cardinal functions?
Thanks!
 A: A necessary condition for $X$ to be 1st countable would that, for all $S \subset X$, sequences suffice to detect points in the closure of $S$. That is, if $f \in \overline S$, then there is a sequence $f_1,f_2,\ldots \in S$ converging to $f$. 
For any finite set $F \subset [0,1]$, it should be possible to construct a continuous function $f_F$ such that $f_F(x) = 0$ iff $x \in F$, and $f_F(\frac{x+y}{2}) = 1$ when $x,y \in F$ are distinct. Let $S \subset X$ be the set of all such functions $f_F$. Is the constant zero function in the closure of $S$? Can it be the limit of a sequence in $S$?
As for (4), it is in fact true that $X$ is still separable when considered under the finer topology of uniform convergence. I would recommend googling the Weierstrass approximation theorem for ideas on what sorts of families of functions you might try for your countable dense set (you won't need to prove this theorem though :p).
For (5), ask yourself the question: "what can I say about about $f,g \in X$ if $f(q) = g(q)$ at every rational number $q \in [0,1]$? Can I enforce this condition on $g$ by requiring it to be in a countable collection of neighbourhoods of $f$?"
