Abstract
Since its first appearence in the book Vorstudien zur Topologie by Johann Benedict Listing of 1847, topology (then and for a long period termed analysis situs ) has been given many facets; among the main ones are considerations of neighborhoods, open sets, and closed sets. We start here, giving the corresponding definitions lifted to point-free as well as quantifier-free versions, showing how they are interrelated, thus exhibiting their cryptomorphism and offering the possibility to transform one version into the other, not least visualizing them via TituRel programs.
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Citation: Es mag erlaubt sein, für diese Art Untersuchungen räumlicher Gebilde den Namen “Topologie” zu gebrauchen statt der von Leibniz vorgeschlagenen Benennung “geometria situs”, welche an den Begriff des Maßes, der hier ganz untergeordnet ist, erinnert, und mit dem bereits für eine andere Art geometrischer Betrachtungen gebräuchlich gewordenen Namen “géométrie de position” collidiert.
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One should not attempt to find a person named Boto von Querenburg! This is just the name of a community of authors at Bochum University working on Topology, situated in the suburb of Querenburg. They provided an influential text, but—sadly—starting with metric spaces, as opposed to our relational approach.
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.References
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Schmidt, G., Winter, M. (2018). Applying Relations in Topology. In: Relational Topology. Lecture Notes in Mathematics, vol 2208. Springer, Cham. https://doi.org/10.1007/978-3-319-74451-3_5
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DOI: https://doi.org/10.1007/978-3-319-74451-3_5
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