The 21 cm forest, a series of narrow absorption features produced by neutral hydrogen against high-redshift radio sources, provides a unique probe of the thermal and small-scale structure of the early Universe. However, X-ray heating and dark-matter free streaming can both suppress small-scale absorption features, leading to degeneracies in conventional statistics.

In this talk, I will introduce the basic physics of the 21 cm forest and present a new approach based on topological data analysis. Using persistent homology, we characterize how absorption troughs appear, persist, and merge, and construct topological descriptors sensitive to heating and warm dark matter. We show that combining these descriptors helps break their degeneracy, while retaining useful information even in the presence of SKA1-Low-like thermal noise.