Question: Show that there exists a Locally Compact, Perfect set in $[0,1]$ such that ternary expansion of each of its points consists of $0$ and $1$ only.
I know that the Cantor Set has almost all the above properties except the last one.
$1.$ Cantor set is compact and hence locally compact.
$2.$ Cantor set is a perfect set.
$3.$ Cantor set consists of members whose ternary expansion consists of $0$ and $2$ only. But here we required $0$ and $1$ only. How should I proceed? Thanks in advance!