Topologies conceptual confusion (topology of maximum norm/of pointwise convergence) One of the questions from my lectures notes reads as follows:
"Show that the identity map from $C[0,1]$ with the topology, $T_m$, induced by the maximum norm to the topology of pointwise convergence, $T_p$, is continuous."
The problem is I don't know what either of these topologies are exactly as it isn't explained at all in my notes; I have been told that the topology of pointwise convergence on the set of real functions $F[X]$ is defined as a topology with the sub basis formed by the sets $\{f \in F[X] | a < f(x) < b\}(x \in X, a,b \in \mathbb{R})$ but I don't even understand what that means at all. Could somebody try to briefly guide me as to what these topologies mean so that I can try to address the question at hand? 
Thanks so much in advance.
 A: *

*Topology of point-wise convergence: this is precisely the so called product topology on the set $\mathbb{R}^{[0,1]}$, i.e. all functions from $[0,1]$ to $\mathbb{R}$. We simply restrict it to the subset $C[0,1]$. If you're not familiar with the concept of product topology, then this is what you can conclude from the information that you have been given. For each $x\in X$ and $a,b\in\mathbb{R}$ you look at the set $$O_{x,a,b}:=\{f\in F[X]:a<f(x)<b\},$$
i.e. all functions that take value in the interval $(a,b)$ at the fixed point $x$. So this is a collection of functions on $X$. We use this as the sub-basis for the topology, whence the basis is a collection of finite intersection of sets of the type $O_{x,a,b}$ where we let $x\in X$ and $a,b\in\mathbb{R}$ vary. So the basis sets are of the form
$$O_{(x_{1},...,x_{n}),(a_{1},...,a_{n}),(b_{1},...,b_{n})}:= \{f\in F[X]:a_{i}<f(x_{i})<b_{i}\;\forall i\},$$
i.e. the collection of all functions that at points $x_{i}$ take values in the intervals $(a_{i},b_{i})$. Draw a couple of pictures and try to see how this makes up a topology. This is called the product topology.

*The topology induced by the maximum norm: this is the sup-topology. We have the norm $$\|f-g\|=\max_{x\in [0,1]}|f(x)-g(x)|$$ defined on $C[0,1]$ that induces our topology. Note that the maximum indeed exists because continuous functions attain extremums on compact sets. So this topology has open balls as basis elements (norm spaces are metric spaces). An arbitrary basis element would be
$$B(f,\varepsilon)=\{g\in C[0,1]:\|f-g\|<\varepsilon\},$$
for some $f\in C[0,1]$ and $\varepsilon>0$. This is a collection of all functions that lie in the $\varepsilon$-tube around $f$.
I hope this helps you to get started with the problem.
