General topology – Continuum fractal plane with the command \$ omega \$?

continuum means compact and connected.

L & # 39; s order $$ord (x)$$ from one point $$x$$ in a continuum $$X$$ is defined to be the least ordinal $$alpha$$ such as $$X$$ has a gaming neighborhood base open to $$x$$ without more than $$alpha$$ points their limits.

The triangle of Sierpinski has three points of order $$2$$, many motions of order $$4$$ (the tops of the other triangles), and all the other points are of order $$3$$.

The Sierpinski carpet is in order $$mathfrak c = | mathbb R |$$ at each of his points.

I am looking for something to do between the Sierpinski triangle and the Sierpinski carpet.

Question. Is there a fractal plane continuum that is in order $$omega$$ at each of his points?

fractal One can vaguely interpret "self-similar" or "simple recursive construction".