Abstract: A function's excursion set is the sub-domain where its value exceeds some threshold. Some key examples illustrating the central role that excursion sets play in different application areas are as follows. In medical imaging, in order to understand the brain parts involved in a particular task, analysts frequently look at the high blood flow level regions in the brain when the said task is being performed. In control theory, it is known that the viability and invariance properties of control systems can be expressed as super-level sets of suitable value functions. In robotics, in order to plan its motion, a sensor robot may want to identify the sub-terrain where the accessibility probability is above some threshold. Often, functions whose excursions are of interest are either random themselves (for e.g., due to noise) or, while being deterministic, are too complicated and hence can be treated as being a sample of a random field. In this sense, studying the topology of random field excursions is vital. This work is the first detailed study of their Betti numbers (number of holes) in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further have a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools (joint work with Sunder Ram Krishnan).