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.

enter the description of the image here

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".