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We introduce a canonical, compact topology, which we call weakly causal,naturally generated by the causal site of J. D. Christensen and L. Crane, apointless algebraic structure motivated by certain problems of quantum gravity.We show that for every four-dimensional globally hyperbolic Lorentzian manifoldthere exists an associated causal site, whose weakly causal topology isco-compact with respect to the manifold topology and vice versa. Thus, thecausal site has the full information about the topology of space-time,represented by the Lorentzian manifold. In addition, we show that there existalso non-Lorentzian causal sites (whose causal relation is not a continuousposet) and so the weakly causal topology and its de Groot dual extends theusual manifold topology of space-time beyond topologies generated by thetraditional, smooth model. As a source of inspiration in topologizing thestudied causal structures, we use some methods and constructions of generaltopology and formal concept analysis.

The Framework, Causal and Co-compact Structure of Space-time