Continuum means compact and connected.
Define the order of a point $x$ in a continuum $X$ to be the least cardinal $\alpha$ such that $X$ has a neighborhood base of open sets at $x$ with no more than $\alpha$ points their boundaries.
The Sierpinski triangle has three points of order $2$, countably many points of order $4$ (the vertices of the other triangles), and all other points are of order $3$.
The Sierpinski carpet has order $\mathfrak c=|\mathbb R|$ at each of its points.
I am looking for something to go between the Sierpinski triangle and the Sierpinski carpet.
Question 1. Is there a fractal plane continuum which has order $\aleph_0$ at each of its points?
Fractal can be loosely interpreted here to mean "self-similar", "simple recursive construction", or "intersection of an easily definable nested sequence of plane domains".
A slight variation on Question 1:
Question 2. Is there a fractal plane continuum which has a basis of open sets with countably infinite boundaries?