On complete metric spaces containing the Sierpinski curve

It is proved that a complete metric space topologically contains the Sierpinski universal plane curve if and only if it has a subset with so-called bypass property, i.e. it has a subset $K$ containing an arc such that for each $a\in K$ and for each open arc $A\subset K$ with $a\in A$, there exists an arbitrary small arc in $K\setminus \{a\}$ joining the two components of $A\setminus \{a\}$.