Connected and compact subset of $R^2$ not a smooth manifold and smooth manifold w/ boundary in $R^2$

Question

Give an example of a connected, compact subset of $R^2$ that is neither a smooth manifold nor a manifold with boundary.

Appreciate any help!

Answer 1

Hint for this question: 2 things that can "go wrong" (choose your favorite).

1) Your "manifold" doesn't have a consistent dimension.

2) You can't define a tangent plane/line at a point in the set.

In this way it should be possible to adapt any old example that you already know of sets that are not manifolds.