The shark teeth is a topological IFS-attractor

TL;DR

This paper demonstrates that the 'shark teeth' space is a topological IFS-attractor, expanding understanding of fractal-like structures beyond traditional IFS-attractors and showing such spaces can differ topologically.

Contribution

It introduces the concept of topological IFS-attractors and provides an example of a space that is a topological IFS-attractor but not homeomorphic to any standard IFS-attractor.

Findings

Shark teeth space is a topological IFS-attractor.

Existence of a space that is a topological IFS-attractor but not a traditional IFS-attractor.

Provides a new perspective on fractal structures in topology.

Abstract

We show that the space called shark teeth is a topological IFS-attractor, that is for every open cover of $X=\bigcup_{i=1}^nf_i(X)$, its image under every suitable large composition from the family of continuous functions $\{f_1,...,f_n\}$ lies in some set from the cover. In particular, there exists a space which is not homeomorphic to any IFS-attractor but is a topological IFS-attractor.