Pointfree topology regards certain order-theoretical structures, called frames, as topological spaces. We regard frames as spaces in virtue of the lattice of open set of every space being a frame. Not all frames are lattices of open sets; nonetheless, studying all frames as spaces offers several advantages, which I will briefly explain. Symmetrically, several spaces might have the same frame of open sets, however there is a certain subcategory of topological spaces which is indeed faithfully extended by the category of frames: that of sober spaces. Sobriety is a property weaker than \( T_2 \) and stronger than \( T_0 \).

We will explore different ways of extending the classical correspondence so that not only sober, but all \( T_0 \) spaces are captured. We will compare three different approaches, all developed independently: (1) that of Raney extensions, which identifies each space with the embedding of the open sets into the complete lattice of all their intersections; (2) that of MT-algebras, which identifies each space with the embedding of the opens into the powerset of the space; (3) that of strictly zero-dimensional biframes, which regards every space as a bitoplogical structure, by considering, next to the original topology, the one generated by the closed sets. I will show that at the level of objects the three approaches are equivalent, show how they do not agree for morphisms, and discuss the desiderata if we are to find a unified approach to pointfree \( T_0 \) spaces.