Inductive dimensions of coarse proximity spaces
Pawel Grzegrzolka, Jeremy Siegert
TL;DR
The work extends Dranishnikov's asymptotic inductive dimension to coarse proximity spaces, establishing $asInd(X)$ as a coarse invariant and relating it to the boundary dimension $Ind(\mathcal{U}X)$. It proves $Ind(\mathcal{U}X)\le asInd(X)$ and shows that $\,\dim(\mathcal{U}X)\le asInd(X)$, including a counterexample where the boundary dimension is strictly smaller, thus negating a naive equality with the boundary. A key contribution is the introduction of complete traceability: if the boundary is completely traceable, then $asInd(X)=Ind(\mathcal{U}X)$. The paper identifies broad classes where this equality holds, notably when the boundary is a $Z$-set or when $X$ admits a metrizable compactification, and it applies the framework to the Freudenthal boundary to relate $asInd(X)$ to $Ind(FX\setminus X)$. These results connect large-scale dimension theory with boundary/topological properties and provide concrete criteria for when asymptotic inductive dimension aligns with boundary inductive dimension.
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.