Hegel and Modern Topology

In this paper we show how some fundamental concepts of modern topology can be understood philosophically in the light of Hegel's Logic as well how modern topological concepts can provide concrete illustrations of many of the concepts and deductions that Hegel used. This paper can be seen as a continuation of our paper \cite{pro} where we argued that the prototypes of many fundamental notions of modern topology were already found in Aristotle's Physics. This paper is divided into two parts. The first part (sections 2 - 9) focuses only on elementary topology while the remaining sections discuss advanced category theory. Update (July 2025). Since this note was written the author has had occasion to deepen their understanding of these matters (in particular during reading groups on Hegel's Phenomenology and Spirit and Science of Logic) and it is their view that there is now much to be both corrected and added. In particular some very interesting connections to the philosophy of Proclus (and in particular the theory of dialectics and the Proclean view of mathematics in his Commentary of the the first book of the Elements). Also we recommend the very interesting PhD thesis by Fabio Mascarenhas Nolasco, A suspensão qualitativa da quantidade: a crítica de Hegel ao paradigma matemático da ciência moderna (2015). All this would require a radical rewriting of the entire paper. Instead of this we must be content for now with merely presenting some additions at the beginning.