On $T_1$-reflection of topological spaces

Year 2023, Volume: 52 Issue: 2, 398 - 409, 31.03.2023

Sami Lazaar , Abdelwaheb Mhemdi , Tareq Al-shami , Hadjer Okbani

https://doi.org/10.15672/hujms.1121712

Abstract

This paper deals with some universal spaces. For every topological space $X$, the universal $T_1$ space is viewed as the bottom element of the lattice $\mathcal{L}_X$. The class of morphisms in $\mathrm{\mathbf{Top}}$ orthogonal to all $T_1$ spaces is characterized. Also, we introduce some new separation axioms and characterize them. Moreover, we characterize topological spaces $X$ for which the universal $T_1$ space associated with $X$ is a spectral space. Finally, we give some characterizations of topological spaces such that their $T_1$-reflection are compact spaces.

Keywords

$T_1$-spaces, separation axioms, lattice, spectral spaces

References

• [1] A. Ayache and O. Echi, The envelope of a subcategory in Topology and group theory, Int. J. Math. Math. Sci. 21, 3787–3404, 2005.
• [2] K. Belaid, H-spectral spaces, Topol. Appl. 153, 3019–3023, 2006.
• [3] K. Belaid, O. Echi, and R. Gargouri, A-spectral spaces, Topol. Appl., 138, 315–322, 2004.
• [4] K. Belaid, O. Echi, and S. Lazaar, $T_{(\alpha , \beta )}$-spaces and the Wallman compactification, Int. J. Math. Math. Sci. 68, 3717–3735, 2004.
• [5] E. Bouacida, O. Echi, and E. Salhi, Foliations, spectral topology and special morphisms, Lect. Notes in Pure and Appl. Math. 205, 111–132, 1999.
• [6] E. Bouacida, O. Echi, and E. Salhi, Feuilletages et topologie spectrale, J. Math. Soc. Jpn. 52 (2), 447–464, 2000.
• [7] O. Bratteli and G.A. Elliott. Structure spaces of approximately finite-dimensional $C^{\star}$-algebras II, J. Funct. Anal. 30 (1), 74–82, 1978.
• [8] C. Casacuberta, A. Frei, and G. C. Tan, Extending localization functors, J. Pure Appl. Algebra 103, 149–165, 1995.
• [9] A. Deleanu, A. Frei, and P. Hilton, Generalized Adams completion, Cah. Topologie Géom. Différ. Catég. 15, 61–82, 1974.
• [10] D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, Kluwer Academic Publishers, 1995.
• [11] R. El Bashir and J. Velebil, Simultaneously reflective and coreflective subcategories of presheaves, Theory Appl. Categ. 10, 410–423, 2002.
• [12] A. Frei, On completion and shape, Bol. Soc. Brasil. Mat., 5, 147–159, 1974.
• [13] P. J. Freyd and G. M. Kelly, Categories of continuous functors (I), J. Pure Appl. Algebra 2, 169–191, 1972.
• [14] A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, Die Grundlehren der Mathematischen Wissenschaften 166, Springer-Verlag, New York, 1971.
• [15] J. Hartmanis, On the lattice of topologies, Canad. J. Math. 10, 547–553, 1958.
• [16] J. M. Harvey, Reflective subcategories Ill, J. Math. 29, 365–369, 1985.
• [17] H. Herrlich and G. Strecker, H-closed spaces and reflective subcategories, Math. Ann. 177, 302–309, 1968.
• [18] M. Hochster, Prime ideal structure in commutative rings, Trans. Am. Math. Soc. 142, 43–60, 1969.
• [19] M. Lamper, Complements in the lattice of all topologies of topological groups, Arch Math. (Brno) 10 (4), 221–230, 1974.
• [20] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math. 5, Springer-Verlag, New York, 1971.
• [21] W. Tholen, Reflective subcategories, Topol. Appl. 27, 201–212, 1987.
• [22] W.J. Thron, Lattice-equivalence of topological spaces, Duke. Math. J. 29, 671–679, 1962.