# Is each Peano continuum a topological fractal?

Problem. Is each Peano continuum a topological fractal?

A compact Hausdorff space $$X$$ is a topological fractal if $$X=\bigcup_{i=1}^n f_i(X)$$ for some continuous maps $$f_1,\dots,f_n:X\to X$$ such that for any sequence $$(g_i)_{i\in\omega}\subset\{f_1,\dots,f_n\}^{\omega}$$ the intersection $$\bigcap_{m\in\omega} g_0\circ\dots\circ g_m(X)$$ is a singleton.

(The problem was posed on 16.11.2014 by Taras Banakh on page 3 of Volume 0 of the Lviv Scottish Book).

The prize for solution: A lunch in "Szkocka".