A Regular non-Tychonoff space compactificable by N

  by   Martin Kovár, Alena Chernikava

Abstract and usage:

Two topological spaces X, Y are mutually compactificable if there exists a compact (not necessarily Hausdorff) topology on their disjoint union, which induces on both of the sets X,Y their original topologies, such that any two points, on taken from X and the other from Y, have disjoint neighborhoods. The problem how to determine, whether two topological spaces are mutually compactificable is so far unsolved in the most general case. It was proved by the first author that if two spaces constitute a mutually compactificable pair, both are -regular. The application Compactificability Simulator simulates the topological properties of a space, defined by J. Thomas already in 1969. The space, which is, of course, infinite, is simulated by a sequence of finite approximations, parametrized by two parametres, available for the user's input. The selected space is interesting since it is regular, but not Tychonoff (such examples are relatively rare in topology). The fact that the space of J. Thomas is also mutually compactificable with a countably infinite discrete space was proved by the first autor in 2007. The authors hope that the aplication could help to more deeply understand to the topological proceses in the neighborhoods of the points of the both spaces, the space of J. Thomas and the countably infinite discrete space, which are responsible for the mutual compactificability. For details and the construction of the studied topology we refer the reader to [1]. The reseach has also some potential impact to the study of the singularity properties and asymptotical behavior of the integral curves of various systems of differential equations.

  Bow number:  
  Iteration parameter:  


Iteration of the selected segment of the studied space:


The blue points represent the elements of the countably infinite discrete topological space which compactificates (and so it is mutually compactificable with) the constructed space which is the subject of the simulation. The studied space is topologically represented by the iterations as their inverse limit. It also should be noted that the studied topology is not Euclidean (and even not Hausdorff), so the fact that the "bows" converge in some sense to the blue points can not be observed taking the usual topology induced from the real plane.


The application Compactificability Simulator is written in Java powered by Wolfram webMathematica 3.0. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. Its earlier versions were presented by the authors on several scientific conferences, like Aplimat and International Colloquium in Brno, as an integral part of their research. In case of interest in more theoretical background, see [1] or contact the authors. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-11-2/921 "Vlastnosti řešení funkcionálních diferenciálních a diferenčních rovnic"

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