Fractals are ubiquitous in nature, and since Mandelbrot's seminal insight into their structure, there has been growing interest in them. While the topological properties of the limit sets of IFSs have been studied -- notably in the pioneering work of Hata -- many aspects remain poorly understood, especially in the non-autonomous setting. In this paper, we present a homological framework which captures the structure of the limit set. We apply our novel abstract theory to the concrete analysis of the so-called fractal square, and provide an answer to a variant of Mandelbrot's percolation problem. This work offers new insights into the topology of fractals.