Is every connected subset of the Sierpiński triangle arcwise connected?

Solution 1:

Perhaps surprisingly, this is false. And it's quite an old result of Knaster and Kuratowski: http://www.ams.org/journals/bull/1927-33-01/S0002-9904-1927-04326-9/S0002-9904-1927-04326-9.pdf

In short, let $V$ be the set of vertices of all triangles appearing in the standard construction of the Sierpiński triangle. And let $B$ be any Bernstein subset of the Sierpiński triangle, i.e. a set intersecting, but not containing, every perfect subset of the triangle. Transfinite recursion provides an easy construction since every perfect subset has the cardinality of the whole triangle.

It's easy to see that $B\cup V$ contains no arc. It turns out it's connected, which is the hard part.