# On some generic small Cantor spaces

## TL;DR

This paper demonstrates that generically, compact subsets of Euclidean cubes are zero-dimensional Cantor spaces with a stronger property called being strongly microscopic, highlighting typical fractal-like structures in high-dimensional spaces.

## Contribution

It establishes that the typical compact subset in a Euclidean cube is a strongly microscopic zero-dimensional Cantor space, a novel generic property in topology.

## Key findings

- Most compact subsets are zero-dimensional Cantor spaces.
- These sets are strongly microscopic, a stronger form of zero-dimensionality.
- The result applies to all Euclidean cubes of dimension n ≥ 1.

## Abstract

Let $X = [0,1]^{n}$, $n \geq1$. We show that the typical (in the sense of Baire category) compact subset of $X$ is not only a zero dimensional Cantor space but it satisfies the property of being strongly microscopic, which is stronger than being of dimension zero.