5
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.