Abstract
The epireflective subcategories of \(\mathbf{Top}\), that are closed under epimorphic (or bimorphic) images, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \) and \(\mathbf{Top}\). The epireflective subcategories of \(\mathbf{T_2Unif}\), closed under epimorphic images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is compact \(T_2 \} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{T_2Unif}\). The epireflective subcategories of \(\mathbf{Unif}\), closed under epimorphic (or bimorphic) images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{Unif}\). The epireflective subcategories of \(\mathbf{Top}\), that are algebraic categories, are \(\{ X \mid |X| \le 1 \} \), and \(\{ X \mid X\) is indiscrete\(\} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being varietal, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid X\) is compact \(T_2 \} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being algebraic, are \(\{ X \mid X\) is indiscrete\( \} \), and all epireflective subcategories of \(\{ X \mid X\) is compact \(T_2 \} \). Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of \(T_3\) spaces, closed for products, closed subspaces and surjective images.
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Acknowledgments
The author expresses his thanks to D. Petz, for useful conversations on the subject of the paper, and to G. Richter, for kindly having sent him some of his papers. Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant nos. K68398, K75016, K81146.
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Makai, E. Epireflective subcategories of TOP, T\(_2\)UNIF, UNIF, closed under epimorphic images, or being algebraic. Period Math Hung 72, 112–129 (2016). https://doi.org/10.1007/s10998-016-0110-y
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DOI: https://doi.org/10.1007/s10998-016-0110-y
Keywords
- Birkhoff type theorems
- Categories of (\(T_2\)) topological, proximity and uniform spaces
- Epireflective subcategories
- Closedness under products and (closed) subspaces
- Closedness under epimorphic or bimorphic images
- Algebraic subcategories
- Varietal subcategories