# Inductive dimensions of coarse proximity spaces

## TL;DR

This paper extends the concept of asymptotic inductive dimension to coarse proximity spaces, establishing relationships with boundary dimensions and identifying conditions for their equality, with implications for spaces with specific boundary properties.

## Contribution

It introduces the notion of complete traceability for boundaries, linking the asymptotic inductive dimension of coarse proximity spaces to boundary inductive dimensions.

## Key findings

- Asymptotic inductive dimension is greater or equal to boundary inductive dimension.
- Complete traceability ensures equality of dimensions.
- Spaces with Z-set boundaries or metrizable compactifications have traceable boundaries.

## Abstract

In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are $Z$-sets and spaces admitting metrizable compactifications have completely traceable boundaries.