Let Y be a Continuum. Then there is a set countable of arcs
$$\{l_{n}\phantom{a}|\phantom{a}n\in\mathbb{N}\}$$ such that
$$Z=Y\cup{\left(\bigcup_{n\in\mathbb{N}}{l_{n}}\right)}$$ is a Peano continuum.
Anyone know how to start demonstrating that? They recommended me to use Hahn- Mazurkiewicz'Theorem.
Hahn- Mazurkiewicz'Theorem: Let X be a continuum. Then X is a Peano Continuum if, and only if, there is a function $F:[0,1]\rightarrow X$ continuous and onto.
Here's how I would prove it. Any continuum (indeed, any nonempty compact metrizable space) is a quotient of the Cantor set $K$ by a closed equivalence relation. Embedding $K$ in $[0,1]$ as the usual middle thirds Cantor set, $[0,1]$ can be written as a union of $K$ and countably many arcs which are disjoint from $K$ except at their endpoints. So, if $\sim$ is a closed equivalence relation on $K$ and you extend $\sim$ to a closed equivalence relation $\sim'$ on $[0,1]$ which is just equality on $[0,1]\setminus K$, then $[0,1]/{\sim'}$ naturally decomposes as the union of $K/{\sim}$ and countably many arcs. But $[0,1]/{\sim'}$ is a Peano continuum by the Hahn-Mazurkiewicz theorem.