# Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images

## TL;DR

This paper explores the use of Schmidt's game on spaces of compact sets and functions to analyze generic properties related to fractal dimensions, revealing new winning properties that differ from traditional notions like residuality or full measure.

## Contribution

It introduces a novel application of Schmidt's game to the space of compact sets and functions, identifying new generic properties related to fractal dimensions.

## Key findings

- Certain properties are winning for Schmidt's game but are not residual or of full measure.
- The paper demonstrates stark differences between Schmidt's game genericity and classical genericity notions.
- New properties related to fractal dimensions are shown to be Schmidt's game winning.

## Abstract

We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behaviour of objects in a metric space, mostly in the context of fractal dimensions, and the notion of `generic' we adopt is that of being winning for Schmidt's game. We find properties whose corresponding sets are winning for Schmidt's game that are starkly different from previously established, and well-known, properties which are generic in other contexts, such as being residual or of full measure.