On (pre)-approach spaces within convergence approach spaces
Keywords:
convergence approach spaces, pre-approach spaces, approach spaces, closure, closure function, adherence, reflectionAbstract
The purpose of this note is to illustrate a parallel between (pre)topologies when seen among convergence spaces and (pre)approach spaces when seen among convergence approach spaces. This presentation appears to offer a more complete parallel than in the traditional presentation of these approach structures. This sheds some light on the reflector from the category of convergence approach spaces to that of approach spaces and even on the structure of approach spaces as such. This point of view allows for a characterization of approach spaces among convergence approach spaces represented as pointfree convergence frames as in the work of the author in this journal titled "Approach theory and pointfree convergence".
References
J. Adámek, H. Herrlich, and E Strecker. Abstract and Concrete Categories. John Wiley and Sons, Inc., 1990.
M. Ateş and S. Sagıroğlu Peker. The Fell approach structure. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(3):633-649, 2023.
S. Dolecki and F. Mynard. Convergence Foundations of Topology. World Scientific, 2016.
J. Goubault-Larrecq and F. Mynard. Convergence without points. Houston Journal of Mathematics, 46(1):227-282, 2020.
G. Greco Limites et fonctions d'ensemble. Rend. Sem. Mat. Padova, 72:89-97, 1984
Dirk Hofmann, Gavin J Seal, and Walter Tholen. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, volume 153. Cambridge University Press, 2014.
G. Jäger. Convergence approach spaces and approach spaces as lattice-valued convergence spaces. Iranian Journal of Fuzzy Systems, 9(4):1-16, 2012.
G. Jäger. Quantale-valued generalizations of approach spaces: $L$-approach systems. Topology Proceedings, 51:253-276, 2018.
G. Jäger. Quantale-valued generalizations of approach spaces and quantale-valued topological spaces. Quaestiones Mathematicae, 42(10):1313-1333, 2019.
G. Jäger. Quantale-valued convergence tower spaces: Diagonal axioms and continuous extension. Filomat, 35(11):3801-3810, 2021.
H. Lai and W. Tholen. Quantale-valued topological spaces via closure and convergence. Topology and its Applications, 230:599-620, 2017.
E. Lowen and R. Lowen. A quasitopos containing CONV and MET as full subcategories. Int. J. Math. Sci., 19:417-438, 1988.
E. Lowen and R. Lowen. Topological quasitopos hulls of categories containing topological and metric objects. Cahiers de Topologies et Géometrie différentielle Catégorique, 30:213-228, 1989.
E. Lowen, R. Lowen, and C. Verbeeck. Exponential objects in PRAP. Cahiers de Topologies et Géometrie différentielle Catégorique, 38:259-276, 1997.
R. Lowen. Approach spaces: A common supercategory of TOP and MET. Math. Nachr., 141:183-226, 1989.
R. Lowen. Approach Spaces: The Missing Link in Topology-Uniformity-Metric Triad. Oxford University Press, 1997.
R. Lowen. Index Analysis: Approach Theory at Work. Springer Monographs in Mathematics. Springer, 2015.
R. Lowen and M. Sioen. Weak representations of quantified hyperspace structures. Topology and its Applications, 104(1):169-179, 2000.
R. Lowen and P Wuyts. The Vietoris hyperspace structure for approach spaces. Acta Mathematica Hungarica, 139(3):286-302, 2013.
Frédéric Mynard. More pointfree convergence: on convergence frames. in preparation.
Frédéric Mynard. Approach theory and pointfree convergence. Top. Proc., 61(31-47), 2023.
Frédéric Mynard. Measure of compactness for filters in the approach setting. Quaestiones Math., 31:189-201, 2008.
J. Picado and A. Pultr. Frames and locales: Topology without points. Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2012.
