On $T_1$-reflection of topological spaces

Authors : Sami LAZAAR, Abdelwaheb MHEMDİ, Tareq AL-SHAMİ, Hadjer OKBANİ

Pages : 398-409

Doi:10.15672/hujms.1121712

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Publication Date : 2023-03-31

Article Type : Research Paper

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