Is topology on $C[0,1]$ metrizable? Let $C[0,1]$ be the space of all continuous functions on the interval $[0,1]$, then you can induce a topology of pointwise convergence, right? First question, how will these open sets look like? My other question, is this space  metrizable? If not, why? 
 A: Here are some useful results that answer this question and more general questions of the same nature. 
Denote by $C_{p}(X)$ the collection of continuous functions $X\to\mathbb{R}$ with the topology of point wise convergence; define the weight $w(Y)$ of a topological space $Y$ to be the minimum cardinality of a basis; and define the character $\chi(Y):=\sup_{y}\chi(Y,y)$ of a topological space $Y$ to be the supremum over all minimal cardinalities of the neighborhood basis of $Y$, i.e. $\chi(Y,y)$ is the minimum cardinality of a neighborhood basis at $y\in Y$.
Let $Y$ then be an infinite set. 


*

*Theorem #1: we have $|Y|=\chi(C_{p}(Y))=w(C_{p}(Y))$. 

*Theorem #2: the following are equivalent for any infinite cardinal $\kappa$:
\begin{align*}
(i)& \;\; w(Y)\leq \kappa,\\
(ii)&\;\; Y\;\mathrm{embeds}\;\mathrm{in}\;\mathbb{R}^{\kappa},\\
(iii)&\;\; Y\;\mathrm{embeds}\;\mathrm{in}\;[0,1]^{\kappa}.
\end{align*}
Now every metrizable space is first countable, so if $C_{p}(X)$ is metrizable we have by Theorem #1 that $|X|\leq \omega$. And conversely, if $X$ is countable, then by Theorem #1 we have $w(C_{p}(X))=\omega$, and Theorem #2 gives us an embedding $C_{p}(X)\to\mathbb{R}^{\omega}$, making $C_{p}(X)$ metrizable.
Conclusion: $C_{p}(X)$ is metrizable if and only if $X$ countable.
To your specific example, since $[0,1]$ is uncountable, then $C[0,1]$ with point wise convergence is not metrizable.
Reference to Theorems #1 and #2 is e.g. the following book: "V. Tkachuk - $C_{p}$ theory problem book of topological and function spaces", s.209 (p. 165) and s.169 (p. 142).
A: The topology of point-wise convergence in $C[0,1]$ is not metrisable because it is not first-countable. As for the open sets, this topology has a natural subbase consisting of sets of the form
$$\{f\in C[0,1]\colon |f(x)|<\varepsilon\}\quad (x\in [0,1], \varepsilon > 0).$$
