# Embedding topological fractals in universal spaces

### Taras Banakh

Ivan Franko National University, Lviv, Ukraine### Filip Strobin

Lodz University of Technology, Poland

## Abstract

A compact metric space $X$ is called a Rakotch (Banach) fractal if\linebreak $X=\bigcup_{f\in\mathcal F}f(X)$ for some finite system $\mathcal F$ of Rakotch (Banach) contracting self-maps of $X$. A Hausdorff topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\F}f(X)$ for some finite system $\mathcal F$ of continuous self-maps, which is topologically contracting in the sense that for any sequence $(f_n)_{n\in\w}\in\F^\w$ the intersection $\bigcap_{n\in\w}f_0\circ\dots\circ f_n(X)$ is a singleton. It is known that each topological fractal is homeomorphic to a Rakotch fractal. We prove that each Rakotch (Banach) fractal is isometric to the attractor of a Rakotch (Banach) contracting function system on the universal Urysohn space $\mathbb U$. Also we prove that each topological fractal is homemorphic to the attractor $A_\mathcal F$ of a topologically contracting function system $\mathcal F$ on an arbitrary Tychonoff space $U$, which contains a topological copy of the Hilbert cube. If the space $U$ is metrizable, then its topology can be generated by a bounded metric making all maps $f\in\mathcal F$ Rakotch contracting.

## Cite this article

Taras Banakh, Filip Strobin, Embedding topological fractals in universal spaces. J. Fractal Geom. 2 (2015), no. 4 pp. 377–388

DOI 10.4171/JFG/25