Some categorical aspects of the inverse limits in ditopological context
Keywords:inverse limit, natural transformation, co-adjoint functor, ditopology, concrete isomorphism, joint topology
This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits.
J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories (John Wiley & Sons, Inc., 1990). Volume 17, Springer-Verlag, 1990.
L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems 98 (1998), 217-224. https://doi.org/10.1016/S0165-0114(97)00358-8
L. M. Brown, R. Ertürk, Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy Sets and Systems 110, no. 2 (2000), 227-236. https://doi.org/10.1016/S0165-0114(98)00157-2
L. M. Brown, R. Ertürk, S. Dost, Ditopological texture spaces and fuzzy topology, III. Separation Axioms, Fuzzy Sets and Systems 157, no. 14 (2006), 1886-1912. https://doi.org/10.1016/j.fss.2006.02.001
S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton, New Jersey, Princeton University Press, 1952. https://doi.org/10.1515/9781400877492
R. Engelking, General Topology (Helderman Verlag Berlin, 1989).
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A compendium of continuous lattices (Springer-Verlag, Berlin, 1980).
S. Özçag, F. Yildiz and L. M. Brown, Convergence of regular difilters and the completeness of di-uniformities, Hacettepe Journal of Mathematics and Statistics 34 (2005), 53-68.
I. U. Tiryaki and L. M. Brown, Plain ditopological texture spaces, Topology and its Applications 158 (2011), 2005-2015. https://doi.org/10.1016/j.topol.2011.06.051
F. Yildiz and L. M. Brown, Characterizations of real difunctions, Hacettepe Journal of Mathematics and Statistics 35, no. 2 (2006), 189-202.
F. Yildiz and L. M. Brown, Categories of dicompact bi-$T_2$ texture spaces and a Banach-Stone theorem, Quaestiones Mathematicae 30 (2007), 167-192. https://doi.org/10.2989/16073600709486192
F. Yildiz and L. M. Brown, Real dicompact textures, Topology and its Applications 156, no. 11 (2009), 1970-1984. https://doi.org/10.1016/j.topol.2009.03.021
F. Yildiz and L. M. Brown, Dicompleteness and real dicompactness of ditopological texture spaces, Topology and Its Applications 158, no. 15 (2011), 1976-1989. https://doi.org/10.1016/j.topol.2011.06.040
F. Yildiz and L. M. Brown, Extended real dicompactness and an application to Hutton spaces, Fuzzy Sets and Systems 227 (2013), 74-95. https://doi.org/10.1016/j.fss.2013.03.012
F. Yildiz, Connections between real compactifications in various categories, Quaestiones Mathematicae 38, no. 3 (2015),31-455. https://doi.org/10.2989/16073606.2014.981726
F. Yildiz, Completeness types for uniformity theory on textures, Filomat 29, no. 1 (2015), 159-178. https://doi.org/10.2298/FIL1501159Y
F. Yildiz, Inverse systems and inverse limits in the category of plain textures, Topology and Its Applications 201 (2016), 217-234. https://doi.org/10.1016/j.topol.2015.12.038
F. Yildiz, Inverse systems and limits in the category of ditopological plain spaces, Topology and Its Applications 228 (2017), 47-67. https://doi.org/10.1016/j.topol.2017.05.005
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