Every component of a fractal square is a Peano continuum

TL;DR

This paper proves that each component of a fractal square, formed by a specific self-similar construction, is locally connected, highlighting a key topological property of these fractals.

Contribution

It establishes that all components of fractal squares are locally connected, a property not necessarily shared by their three-dimensional analogues.

Findings

Components of fractal squares are locally connected.

Three-dimensional fractal analogues do not share this property.

Provides insight into the topological structure of self-similar sets.

Abstract

This paper concerns the local connectedness of components of self-similar sets. Given an equal partition of the unit square into n*n small squares, we may choose arbitrarily two or more of them and form an iterated function system. The attractor F resulted from this IFS is called a fractal square. We prove that every component of F is locally connected. The same result for three-dimensional analogues of F does not hold.