The classical dual adjunction between frames and spaces, given by the functors \( O\colon \mathbf{Top}\to\mathbf{Frm} \) and \( \mathsf{pt}\colon\mathbf{Frm}\to\mathbf{Top} \), restricts to a dual equivalence between sober spaces and spatial frames. Another pointfree approach to the study of spaces is the so-called \( T_D \) duality, a similar dual adjunction where the fixpoints are the \( T_D \) spaces. The two dualities are incompatible in that the two spectrum functors are different. Recently, a pointfree approach to spaces whose fixpoints are all \( T_0 \) spaces was introduced. The approach is inspired by Raney duality. In this setting, we study the category of Raney extensions and show that we have a dual adjunction extending \( (O,\mathsf{pt}) \) and capturing all \( T_0 \) spaces. By suitably restricting on objects we recover both the classical and the \( T_D \) duality. Other pointfree notions of \( T_0 \) spaces are given by strictly zero-dimensional biframes and McKinsey-Tarski algebras. Aiming to understand the relationship between these three different settings, we introduce and study an abstract notion of pointfree \( T_0 \) space. We study the situation where we have a suitable forgetful functor \( U\colon\mathcal C\to\mathbf{Frm} \) and a dual adjunction \( O_\mathcal C\colon \mathbf{Top}\to\mathcal C \) and \( \mathsf{pt}_\mathcal C\colon \mathbf{Top} \to\mathcal C \) whose fixpoints are the \( T_0 \) spaces and which is, in a sense that will be made precise, compatible with the adjunction \( (O,\mathsf{pt}) \) between frames and spaces. By studying the fibers of \( U \), we are able to study various aspects of \( T_0 \) spaces abstractly. In all pointfree notions of \( T_0 \) spaces mentioned above, the pointfree sober spaces are the top elements of the fibers, and the \( T_D \) ones are the bottom elements.