I have a question regarding Cantor set given to me as a homework question (well, part of it):
a. Prove that the only connected components of Cantor set are the singletons $\{x\}$ where $x\in C$
b. Prove that $C$ is metrizable
I am having some problems with this exercise:
My thoughts about $a$:
I know that in general path connectedness and connectedness are not equivalent, but I know that$\mathbb{R}$ is path connected, I want to say something like that since if $\gamma(t):C\to C$ is continues then $\gamma(t)\equiv x$ for some $x\in C$ then I have it that the connected components of $C$ can be only the singltons.
But I lack any justification - connectedness and path connectedness are not the same thing - but maybe since $\mathbb{R}$ is path connected we can justify somehow that if $C$ had any connected component then it is also path connected ? another thing that confuses me is that the open sets relative to $C$ and relative to $\mathbb{R}$ are not the same so I am also having a problem working with the definition of when a space is called connected
My thoughts about b:
Myabe there is something that I don't understand - but isn't $C$ metrizable since its a subspace of a $[0,1]$ with the topology that comes from the standard metric on $\mathbb{R}$ ?
I would appreciate any explanations and help with this exercise!