Gabrielov–Vorobjov approximation conjecture without separability

Let $S \subset K \subset \mathbb R^n$ be a bounded definable set with a monotone family of compact sets $\{S_\delta\}_{\delta>0}$ satisfying $S=\bigcup_\delta S_\delta$. Suppose that for sufficiently small $\delta$ there are compact definable sets $S_{\delta,\varepsilon}$ increasing in $\varepsilon\in(0,1)$, with $S_\delta=\bigcap_\varepsilon S_{\delta,\varepsilon}$, and suppose that for sufficiently smaller $\delta'$ and every $\varepsilon'>0$ there is an open set $U\subset K$ such that $S_\delta\subset U\subset S_{\delta',\varepsilon'}$. For sufficiently separated parameters $\varepsilon_0\ll\delta_0\ll\varepsilon_1\ll\delta_1\ll\cdots\ll\varepsilon_n\ll\delta_n$, the approximation conjecture without separability. The set

$$S_{\delta_0,\varepsilon_0}\cup\cdots\cup S_{\delta_n,\varepsilon_n}$$

is homotopy equivalent to $S$, without assuming the separability condition. The separability hypothesis is known to hold in many inequality-defined situations but is not preserved under images of definable maps; the conjecture removes that restriction.