Investigation of topological spaces using relators
Keywords:(generalized) uniformities, (generalized) topologies, relators
In this paper, we define uniformities and topologies as relators and show the equivalences of these definitions with the classical ones. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz.
Moreover, we prove implications between important topological properties of relators and disprove others. Finally, we show that our earlier analogous definition [G. Pataki, Investigation of proximal spaces using relators, Axioms 10, no. 3 (2021): 143.] for uniformly and proximally filtered property is equivalent to the topological one.
At the end of this paper, uniformities and topologies are defined in the same way. This will give us new possibilities to compare these and other topological structures.
N. Bourbaki, Topologie Générale, Herman, Paris (1953).
S. A. Davis, Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886-893. https://doi.org/10.1080/00029890.1961.11989785
L. Nachbin, Topology and order, D. Van Nostrand (Princetown, 1965).
H. Nakano and K. Nakano, Connector theory, Pacific J. Math. 56 (1975), 195-213. https://doi.org/10.2140/pjm.1975.56.195
G. Pataki, On the extensions, refinements and modifications of relators, Math. Balk. 15 (2001), 155-186.
G. Pataki, Investigation of proximal spaces using relators, Axioms 10, no. 3 (2021): 143. https://doi.org/10.3390/axioms10030143
G. Pataki and A. Száz, A unified treatment of well-chainedness and connectedness properties, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 19 (2003), 101-166.
W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317. https://doi.org/10.1007/BF01440953
Á. Száz, Basic tools and mild continuities in relator spaces, Acta Math. Hungar. 50 (1987), 177-201. https://doi.org/10.1007/BF01903935
Á. Száz, Directed, topological and transitive relators, Publ. Math. Debrecen 35 (1988), 179-196.
Á. Száz, Relators, Nets and Integrals, Unfinished Doctoral Thesis (1991).
A. Weil, Sur les espaces a structure uniforme at sur la topologie générale, Actualités Sci. Ind. 551, Herman and Cie, Paris, 1937.
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