Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis

Given a definable function $f:S\to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S_t^f\{x\in S:f(x)\le t\}$ for all $t\in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S_t^f$ is right-continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S_{t+\delta }^f$ deformation retracts to $S_t^f$ for all sufficiently small $\delta >0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT.