Definition 1. A continuum is a compact, connected, non-empty metric space.
Definition 2. An arc is any space $X$ homeomorphic to the closed interval $[0, 1].$
Definition 3. A continuum of Peano is a continuum locally connected.
Theorem 1. Let $\mathcal{C}$ a Cantor Set. If $X$ is a compact, non-empty metric space, then there is a function $f \colon\mathcal{C}\to X$ a continuous and surjective function.
Theorem 2 [Hahn - Mazurkiewicz]. Let $X$ be a continuum. Then $X$ is a continuum of Peano if and only if there is a function $f \colon [0,1]\to X$ a continuous and surjective function.
Exercise: Let $Y$ be a continuum. Then there is $\{l_{n}:n\in\mathbb{N}\}$ a countable set of arcs such that $Y\cup\left(\bigcup_{n\in\mathbb{N}}{l_{n}}\right)$ is a continuum of Peano.
My attempt: By Theorem 1, there is a then there is a function $g \colon\mathcal{C}\to Y$ a continuous and surjective function. I know that the complement of the Cantor set is a countable union of open and disjoint intervals, this is, $$[0,1]\setminus{\mathcal{C}}=\bigcup_{n\in\mathbb{N}}{(a_{k},b_{k})},\,a_{k},b_{k}\in \mathcal{C}.$$
My intention is to define a function $f\colon [0,1]\to Y$ given by \begin{equation} f(t)=\left\{\begin{matrix} g(t) & t\in\mathcal{C}\\ h_{i}(t) & t\notin\mathcal{C} \end{matrix}\right. \end{equation}
such that $h_{i}(t)=(1-t)g(a_{i})+tg(b_{i}),\forall\, i\in\mathbb{N}\mbox{ and }t\in(0,1).$
I know that $ g $ is uniformly continuous, since $\mathcal{C}$ is compact. I also know that $ Y $ can be embedded as a closed subspace of the Hilbert's Cube. Although I don't know if $ f $ thus defined is continuous.
I have a question: If Hilbert's cube is convex, does the $ f $ function make sense? Although I don't know if convexity is preserved under continuous functions. I also don't know why Hilbert's cube is convex.
Given the case that that function $f$ makes sense, then who would be the countable set of arcs?
Although I do not know if I will have problems with the distance of those arches. That is, is it necessary to establish an isometry to show that said set of arcs has the same length as the arcs of the set $ [0,1] $? I don't know if that question makes sense.
As you can see, I have many doubts. I hope you can help me with any suggestions please.